License: CC BY 4.0
arXiv:2506.01176v1 [math.PR] 01 Jun 2025

Finite Version of the qπ‘žqitalic_q-Analogue of de Finetti’s Theorem

Adyan Dordzhiev aedordzhiev@gmail.com
Abstract.

Let q∈(0,1)π‘ž01q\in(0,1)italic_q ∈ ( 0 , 1 ). We formulate an asymptotic version of the qπ‘žqitalic_q-analogue of de Finetti’s theorem. Using the convex structure of the space of qπ‘žqitalic_q-exchangeable probability measures, we show that the optimal rate of convergence is of order qnsuperscriptπ‘žπ‘›q^{n}italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT.

1. Introduction

Let S⁒(∞)𝑆S(\infty)italic_S ( ∞ ) denote the group of permutations of the natural numbers that move only finitely many elements. A random sequence X1,X2,X3,…subscript𝑋1subscript𝑋2subscript𝑋3…X_{1},X_{2},X_{3},\ldotsitalic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , … is exchangeable if permuting finitely many indices does not change the law of the sequence. That is, for any finite permutation ΟƒβˆˆS⁒(∞)πœŽπ‘†\sigma\in S(\infty)italic_Οƒ ∈ italic_S ( ∞ ),

(X1,X2,X3,…)⁒=𝑑⁒(Xσ⁒(1),Xσ⁒(2),Xσ⁒(3),…).subscript𝑋1subscript𝑋2subscript𝑋3…𝑑subscriptπ‘‹πœŽ1subscriptπ‘‹πœŽ2subscriptπ‘‹πœŽ3…(X_{1},X_{2},X_{3},\ldots)\overset{d}{=}(X_{\sigma(1)},X_{\sigma(2)},X_{\sigma% (3)},\ldots).( italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , … ) overitalic_d start_ARG = end_ARG ( italic_X start_POSTSUBSCRIPT italic_Οƒ ( 1 ) end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT italic_Οƒ ( 2 ) end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT italic_Οƒ ( 3 ) end_POSTSUBSCRIPT , … ) .

The celebrated de Finetti’s theorem states that an infinite random {0,1}01\{0,1\}{ 0 , 1 }-valued exchangeable sequence is a mixture of i.i.d. Bernoulli sequences. In other words, the space of exchangeable probability measures on {0,1}∞superscript01\{0,1\}^{\infty}{ 0 , 1 } start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT is isomorphic (as a convex set) to the space of all Borel probability measures on [0,1]01[0,1][ 0 , 1 ]. The isomorphism is given by the following formula

(1.0.1) ℙ⁒(X1=1,…,Xk=1,Xk+1=0,…,Xn=0)β‰”βˆ«01pk⁒(1βˆ’p)nβˆ’k⁒μ⁒(d⁒p).≔ℙformulae-sequencesubscript𝑋11…formulae-sequencesubscriptπ‘‹π‘˜1formulae-sequencesubscriptπ‘‹π‘˜10…subscript𝑋𝑛0superscriptsubscript01superscriptπ‘π‘˜superscript1π‘π‘›π‘˜πœ‡π‘‘π‘\mathbb{P}(X_{1}=1,\ldots,X_{k}=1,X_{k+1}=0,\ldots,X_{n}=0)\coloneqq\int_{0}^{% 1}p^{k}(1-p)^{n-k}\mu(dp).roman_β„™ ( italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 , … , italic_X start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = 1 , italic_X start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT = 0 , … , italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 0 ) ≔ ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( 1 - italic_p ) start_POSTSUPERSCRIPT italic_n - italic_k end_POSTSUPERSCRIPT italic_ΞΌ ( italic_d italic_p ) .

De Finetti’s theorem can be extended to more general settings [HS55]. The theorem can be proved by establishing a connection with the Hausdorff moment problem [Fel71]. Another proof can be obtained by the moment method [Kir18]. There is also an alternative approach based on harmonic functions on the Pascal graph [BO16].

In [GO09], [GO10] a deformation of the concept of classical exchangeability was studied.

Definition 1.0.2.

For q>0π‘ž0q>0italic_q > 0, a probability measure β„™β„™\mathbb{P}roman_β„™ on {0,1}∞superscript01\{0,1\}^{\infty}{ 0 , 1 } start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT is qπ‘žqitalic_q-exchangeable if for any Ξ΅1,…,Ξ΅n∈{0,1}∞subscriptπœ€1…subscriptπœ€π‘›superscript01\varepsilon_{1},\ldots,\varepsilon_{n}\in\{0,1\}^{\infty}italic_Ξ΅ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_Ξ΅ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ { 0 , 1 } start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT and elementary transposition (i,i+1)𝑖𝑖1(i,i+1)( italic_i , italic_i + 1 ),

(1.0.3) ℙ⁒(Ξ΅1,…,Ξ΅iβˆ’1,Ξ΅i+1,Ξ΅i,Ξ΅i+2,…,Ξ΅n)=qΞ΅iβˆ’Ξ΅i+1⁒ℙ⁒(Ξ΅1,…,Ξ΅n).β„™subscriptπœ€1…subscriptπœ€π‘–1subscriptπœ€π‘–1subscriptπœ€π‘–subscriptπœ€π‘–2…subscriptπœ€π‘›superscriptπ‘žsubscriptπœ€π‘–subscriptπœ€π‘–1β„™subscriptπœ€1…subscriptπœ€π‘›\mathbb{P}(\varepsilon_{1},\ldots,\varepsilon_{i-1},\varepsilon_{i+1},% \varepsilon_{i},\varepsilon_{i+2},\ldots,\varepsilon_{n})=q^{\varepsilon_{i}-% \varepsilon_{i+1}}\mathbb{P}(\varepsilon_{1},\ldots,\varepsilon_{n}).roman_β„™ ( italic_Ξ΅ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_Ξ΅ start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT , italic_Ξ΅ start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT , italic_Ξ΅ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_Ξ΅ start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT , … , italic_Ξ΅ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = italic_q start_POSTSUPERSCRIPT italic_Ξ΅ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_Ξ΅ start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_β„™ ( italic_Ξ΅ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_Ξ΅ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) .

In other words, each additional inversion introduces an exponential penalty governed by the parameter qπ‘žqitalic_q. For q∈(0,1)π‘ž01q\in(0,1)italic_q ∈ ( 0 , 1 ), a qπ‘žqitalic_q-analogue of de Finetti’s theorem for this type of probability measures has been established in [GO09]. See section  (2.1) for a detailed discussion.

The infinite nature of the phase space {0,1}∞superscript01\{0,1\}^{\infty}{ 0 , 1 } start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT plays a crucial role in both formulations, see the introduction of [DF80] for a counterexample. However, de Finetti’s theorem can also be obtained as a limit of the finite version {0,1}nsuperscript01𝑛\{0,1\}^{n}{ 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT as nβ†’βˆžβ†’π‘›n\to\inftyitalic_n β†’ ∞. It was shown in [DF80] that this convergence occurs at an optimal rate of order 1/n1𝑛1/n1 / italic_n. In this note, we obtain a finite version of the qπ‘žqitalic_q-analogue of de Finetti’s theorem, in the spirit of [DF80], with convergence at the sharp rate of order qnsuperscriptπ‘žπ‘›q^{n}italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT.

Acknowledgments. I am deeply grateful to Grigori Olshanski for suggesting the problem and for his guidance throughout this work.

2. Preliminaries

2.1. qπ‘žqitalic_q-Exchangeability

Assume q∈(0,1)π‘ž01q\in(0,1)italic_q ∈ ( 0 , 1 ). We use the standard notation for the qπ‘žqitalic_q-integer, qπ‘žqitalic_q-factorial, qπ‘žqitalic_q-binomial coefficient, and qπ‘žqitalic_q-Pochhammer symbol, respectively,

[n]≔1βˆ’qn1βˆ’q,[n]!≔[1]β‹…[2],…⋅[n],[nk]≔[n]![k]!⁒[nβˆ’k]!,formulae-sequence≔delimited-[]𝑛1superscriptπ‘žπ‘›1π‘žformulae-sequence≔delimited-[]𝑛⋅delimited-[]1delimited-[]2⋅…delimited-[]𝑛≔FRACOPπ‘›π‘˜delimited-[]𝑛delimited-[]π‘˜delimited-[]π‘›π‘˜[n]\coloneqq\frac{1-q^{n}}{1-q},\;\;[n]!\coloneqq[1]\cdot[2],\ldots\cdot[n],\;% \;\genfrac{[}{]}{0.0pt}{}{n}{k}\coloneqq\dfrac{[n]!}{[k]![n-k]!},[ italic_n ] ≔ divide start_ARG 1 - italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG 1 - italic_q end_ARG , [ italic_n ] ! ≔ [ 1 ] β‹… [ 2 ] , … β‹… [ italic_n ] , [ FRACOP start_ARG italic_n end_ARG start_ARG italic_k end_ARG ] ≔ divide start_ARG [ italic_n ] ! end_ARG start_ARG [ italic_k ] ! [ italic_n - italic_k ] ! end_ARG ,
(x;q)nβ‰”βˆi=0nβˆ’1(1βˆ’x⁒qi),  0β©½n⩽∞,formulae-sequence≔subscriptπ‘₯π‘žπ‘›superscriptsubscriptproduct𝑖0𝑛11π‘₯superscriptπ‘žπ‘–β€„β€„0𝑛(x;q)_{n}\coloneqq\prod_{i=0}^{n-1}(1-xq^{i}),\;\;0\leqslant n\leqslant\infty,( italic_x ; italic_q ) start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ≔ ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT ( 1 - italic_x italic_q start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) , 0 β©½ italic_n β©½ ∞ ,

where (x;q)0≔1≔subscriptπ‘₯π‘ž01(x;q)_{0}\coloneqq 1( italic_x ; italic_q ) start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≔ 1. Since q∈(0,1)π‘ž01q\in(0,1)italic_q ∈ ( 0 , 1 ), the qπ‘žqitalic_q-Pochhammer symbol is well-defined for n=βˆžπ‘›n=\inftyitalic_n = ∞.

For a given finite sequence Ο‰βˆˆ{0,1}nπœ”superscript01𝑛\omega\in\{0,1\}^{n}italic_Ο‰ ∈ { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, consider the number of inversions in Ο‰πœ”\omegaitalic_Ο‰

inv⁒(Ο‰):=#⁒{(i,j):1≀i<j≀n⁒and⁒ωi>Ο‰j}.assigninvπœ”#conditional-set𝑖𝑗1𝑖𝑗𝑛andsubscriptπœ”π‘–subscriptπœ”π‘—\mathrm{inv}(\omega):=\#\{(i,j):1\leq i<j\leq n\ \text{and}\ \omega_{i}>\omega% _{j}\}.roman_inv ( italic_Ο‰ ) := # { ( italic_i , italic_j ) : 1 ≀ italic_i < italic_j ≀ italic_n and italic_Ο‰ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT > italic_Ο‰ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT } .

Denote by Cn,k:={Ο‰βˆˆ{0,1}n:βˆ‘i=1nΟ‰i=k}assignsubscriptπΆπ‘›π‘˜conditional-setπœ”superscript01𝑛superscriptsubscript𝑖1𝑛subscriptπœ”π‘–π‘˜C_{n,k}:=\{\omega\in\{0,1\}^{n}:\sum_{i=1}^{n}\omega_{i}=k\}italic_C start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT := { italic_Ο‰ ∈ { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT : βˆ‘ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_Ο‰ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_k } the set of all binary sequences of length n𝑛nitalic_n containing exactly kπ‘˜kitalic_k ones. Consider the sequence sn,k:=(1,1,…,1,0,0,…,0)assignsubscriptπ‘ π‘›π‘˜11…100…0s_{n,k}:=(1,1,\ldots,1,0,0,\ldots,0)italic_s start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT := ( 1 , 1 , … , 1 , 0 , 0 , … , 0 ) in Cn,ksubscriptπΆπ‘›π‘˜C_{n,k}italic_C start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT, which has ones in the first kπ‘˜kitalic_k positions and zeros in the remaining nβˆ’kπ‘›π‘˜n-kitalic_n - italic_k positions. This sequence has the largest number of inversions in Cn,ksubscriptπΆπ‘›π‘˜C_{n,k}italic_C start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT. Each qπ‘žqitalic_q-exchangeable measure β„™β„™\mathbb{P}roman_β„™ on {0,1}nsuperscript01𝑛\{0,1\}^{n}{ 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT is defined by the following equation

(2.1.1) ℙ⁒(Οƒβ‹…Ο‰)=qinv⁒(Ο‰)βˆ’inv⁒(Οƒβ‹…Ο‰)⁒ℙ⁒(Ο‰),Ο‰βˆˆ{0,1}n,ΟƒβˆˆS⁒(n).formulae-sequenceβ„™β‹…πœŽπœ”superscriptπ‘žinvπœ”invβ‹…πœŽπœ”β„™πœ”formulae-sequenceπœ”superscript01π‘›πœŽπ‘†π‘›{\mathbb{P}}(\sigma\cdot\omega)=q^{\text{inv}(\omega)-\text{inv}(\sigma\cdot% \omega)}{\mathbb{P}}(\omega),\quad\omega\in\{0,1\}^{n},\sigma\in S(n).roman_β„™ ( italic_Οƒ β‹… italic_Ο‰ ) = italic_q start_POSTSUPERSCRIPT inv ( italic_Ο‰ ) - inv ( italic_Οƒ β‹… italic_Ο‰ ) end_POSTSUPERSCRIPT roman_β„™ ( italic_Ο‰ ) , italic_Ο‰ ∈ { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_Οƒ ∈ italic_S ( italic_n ) .

In particular, each qπ‘žqitalic_q-exchangeable measure on {0,1}nsuperscript01𝑛\{0,1\}^{n}{ 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT is determined by its values on the family of sequences {sn,k}n,ksubscriptsubscriptπ‘ π‘›π‘˜π‘›π‘˜\{s_{n,k}\}_{n,k}{ italic_s start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT, since

(2.1.2) ℙ⁒(Ο‰)=qinv⁒(Ο‰)⁒ℙ⁒(sn,k),Ο‰βˆˆCn,k.formulae-sequenceβ„™πœ”superscriptπ‘žinvπœ”β„™subscriptπ‘ π‘›π‘˜πœ”subscriptπΆπ‘›π‘˜\mathbb{P}(\omega)=q^{\mathrm{inv}(\omega)}\,\mathbb{P}(s_{n,k}),\quad\omega% \in C_{n,k}.roman_β„™ ( italic_Ο‰ ) = italic_q start_POSTSUPERSCRIPT roman_inv ( italic_Ο‰ ) end_POSTSUPERSCRIPT roman_β„™ ( italic_s start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT ) , italic_Ο‰ ∈ italic_C start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT .

Note that equation (2.1.1) can be extended to the case where Ο‰βˆˆ{0,1}βˆžπœ”superscript01\omega\in\{0,1\}^{\infty}italic_Ο‰ ∈ { 0 , 1 } start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT and ΟƒβˆˆS⁒(∞)πœŽπ‘†\sigma\in S(\infty)italic_Οƒ ∈ italic_S ( ∞ ). It is still equivalent to (1.0.3), since the difference inv⁒(Ο‰)βˆ’inv⁒(Οƒβ‹…Ο‰)invπœ”invβ‹…πœŽπœ”\mathrm{inv}(\omega)-\mathrm{inv}(\sigma\cdot\omega)roman_inv ( italic_Ο‰ ) - roman_inv ( italic_Οƒ β‹… italic_Ο‰ ) is finite whenever ΟƒβˆˆS⁒(∞)πœŽπ‘†\sigma\in S(\infty)italic_Οƒ ∈ italic_S ( ∞ ).

We now prove a useful property of the function inv⁒(Ο‰)invπœ”\text{inv}(\omega)inv ( italic_Ο‰ ).

Proposition 2.1.3.
(2.1.4) βˆ‘Ο‰βˆˆCn,kqinv⁒(Ο‰)=[nk].subscriptπœ”subscriptπΆπ‘›π‘˜superscriptπ‘žinvπœ”FRACOPπ‘›π‘˜\sum_{\omega\in C_{n,k}}q^{\text{inv}(\omega)}=\genfrac{[}{]}{0.0pt}{}{n}{k}.βˆ‘ start_POSTSUBSCRIPT italic_Ο‰ ∈ italic_C start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT inv ( italic_Ο‰ ) end_POSTSUPERSCRIPT = [ FRACOP start_ARG italic_n end_ARG start_ARG italic_k end_ARG ] .
Proof.

The qπ‘žqitalic_q-binomial coefficient is uniquely determined by the following recurrence relation

[nk]=qk⁒[nβˆ’1k]+[nβˆ’1kβˆ’1].FRACOPπ‘›π‘˜superscriptπ‘žπ‘˜FRACOP𝑛1π‘˜FRACOP𝑛1π‘˜1\genfrac{[}{]}{0.0pt}{}{n}{k}=q^{k}\genfrac{[}{]}{0.0pt}{}{n-1}{k}+\genfrac{[}% {]}{0.0pt}{}{n-1}{k-1}.[ FRACOP start_ARG italic_n end_ARG start_ARG italic_k end_ARG ] = italic_q start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT [ FRACOP start_ARG italic_n - 1 end_ARG start_ARG italic_k end_ARG ] + [ FRACOP start_ARG italic_n - 1 end_ARG start_ARG italic_k - 1 end_ARG ] .

By forgetting the last entry in each sequence, we obtain the decomposition

Cn,k=Cnβˆ’1,kβˆ’1βŠ”Cnβˆ’1,k,subscriptπΆπ‘›π‘˜square-unionsubscript𝐢𝑛1π‘˜1subscript𝐢𝑛1π‘˜C_{n,k}=C_{n-1,k-1}\sqcup C_{n-1,k},italic_C start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT = italic_C start_POSTSUBSCRIPT italic_n - 1 , italic_k - 1 end_POSTSUBSCRIPT βŠ” italic_C start_POSTSUBSCRIPT italic_n - 1 , italic_k end_POSTSUBSCRIPT ,

and as a result, we have

βˆ‘Ο‰βˆˆCn,kqinv⁒(Ο‰)=qkβ’βˆ‘Ο‰βˆˆCnβˆ’1,kqinv⁒(Ο‰)+βˆ‘Ο‰βˆˆCnβˆ’1,kβˆ’1qinv⁒(Ο‰).subscriptπœ”subscriptπΆπ‘›π‘˜superscriptπ‘žinvπœ”superscriptπ‘žπ‘˜subscriptπœ”subscript𝐢𝑛1π‘˜superscriptπ‘žinvπœ”subscriptπœ”subscript𝐢𝑛1π‘˜1superscriptπ‘žinvπœ”\sum_{\omega\in C_{n,k}}q^{\mathrm{inv}(\omega)}=q^{k}\sum_{\omega\in C_{n-1,k% }}q^{\mathrm{inv}(\omega)}+\sum_{\omega\in C_{n-1,k-1}}q^{\mathrm{inv}(\omega)}.βˆ‘ start_POSTSUBSCRIPT italic_Ο‰ ∈ italic_C start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT roman_inv ( italic_Ο‰ ) end_POSTSUPERSCRIPT = italic_q start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT βˆ‘ start_POSTSUBSCRIPT italic_Ο‰ ∈ italic_C start_POSTSUBSCRIPT italic_n - 1 , italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT roman_inv ( italic_Ο‰ ) end_POSTSUPERSCRIPT + βˆ‘ start_POSTSUBSCRIPT italic_Ο‰ ∈ italic_C start_POSTSUBSCRIPT italic_n - 1 , italic_k - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT roman_inv ( italic_Ο‰ ) end_POSTSUPERSCRIPT .

This identity coincides with the recurrence relation defining the qπ‘žqitalic_q-binomial coefficient.

∎

In [GO09], a qπ‘žqitalic_q-analogue of de Finetti’s theorem was established. Consider the qπ‘žqitalic_q-analogue of the interval [0,1]01[0,1][ 0 , 1 ]

Ξ”q≔{1,q,q2,…}βˆͺ{0}.≔subscriptΞ”π‘ž1π‘žsuperscriptπ‘ž2…0\Delta_{q}\coloneqq\{1,q,q^{2},\ldots\}\cup\{0\}.roman_Ξ” start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ≔ { 1 , italic_q , italic_q start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , … } βˆͺ { 0 } .

For each xβˆˆΞ”qπ‘₯subscriptΞ”π‘žx\in\Delta_{q}italic_x ∈ roman_Ξ” start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT, we define a qπ‘žqitalic_q-analogue of the Bernoulli measure Ξ½xqsubscriptsuperscriptπœˆπ‘žπ‘₯\nu^{q}_{x}italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT on {0,1}∞superscript01\{0,1\}^{\infty}{ 0 , 1 } start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT and {0,1}nsuperscript01𝑛\{0,1\}^{n}{ 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT as the unique qπ‘žqitalic_q-exchangeable measure whose values on standard cylinder sets are assigned according to the formula

(2.1.5) Ξ½xq⁒(X1=1,…,Xk=1,Xk+1=0,…,Xn=0)≔qβˆ’k⁒(nβˆ’k)⁒xnβˆ’k⁒(x;qβˆ’1)k.≔subscriptsuperscriptπœˆπ‘žπ‘₯formulae-sequencesubscript𝑋11…formulae-sequencesubscriptπ‘‹π‘˜1formulae-sequencesubscriptπ‘‹π‘˜10…subscript𝑋𝑛0superscriptπ‘žπ‘˜π‘›π‘˜superscriptπ‘₯π‘›π‘˜subscriptπ‘₯superscriptπ‘ž1π‘˜\nu^{q}_{x}(X_{1}=1,\ldots,X_{k}=1,X_{k+1}=0,\ldots,X_{n}=0)\coloneqq q^{-k(n-% k)}x^{n-k}(x;q^{-1})_{k}.italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 , … , italic_X start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = 1 , italic_X start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT = 0 , … , italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 0 ) ≔ italic_q start_POSTSUPERSCRIPT - italic_k ( italic_n - italic_k ) end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT italic_n - italic_k end_POSTSUPERSCRIPT ( italic_x ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT .

Interpreting xβˆˆΞ”qπ‘₯subscriptΞ”π‘žx\in\Delta_{q}italic_x ∈ roman_Ξ” start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT as the probability of a zero, the polynomial defined in  (2.1.5) plays the role of a qπ‘žqitalic_q-analogue for the binomial term xk⁒(1βˆ’x)nβˆ’ksuperscriptπ‘₯π‘˜superscript1π‘₯π‘›π‘˜x^{k}(1-x)^{n-k}italic_x start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( 1 - italic_x ) start_POSTSUPERSCRIPT italic_n - italic_k end_POSTSUPERSCRIPT.

Theorem 2.1.6.

(Gnedin-Olshanski) qπ‘žqitalic_q-exchangeable probability measures on {0,1}∞superscript01\{0,1\}^{\infty}{ 0 , 1 } start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT are in one-to-one correspondence with probability measures on Ξ”qsubscriptΞ”π‘ž\Delta_{q}roman_Ξ” start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT. The bijection has the form

(2.1.7) β„™β‰”βˆ«Ξ”qΞ½xq⁒μ⁒(d⁒x),≔ℙsubscriptsubscriptΞ”π‘žsubscriptsuperscriptπœˆπ‘žπ‘₯πœ‡π‘‘π‘₯\mathbb{P}\coloneqq\int_{\Delta_{q}}\nu^{q}_{x}\,\mu(dx),roman_β„™ ≔ ∫ start_POSTSUBSCRIPT roman_Ξ” start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_ΞΌ ( italic_d italic_x ) ,

The classical version corresponds to the limit qβ†’1β†’π‘ž1q\to 1italic_q β†’ 1. As qπ‘žqitalic_q increases, the set Ξ”qsubscriptΞ”π‘ž\Delta_{q}roman_Ξ” start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT becomes denser, and at q=1π‘ž1q=1italic_q = 1, it fills the entire interval [0,1]01[0,1][ 0 , 1 ].

2.2. Finite form of classical version

We recall the main result from [DF80]. Given a probability measure ΞΌπœ‡\muitalic_ΞΌ on [0,1]01[0,1][ 0 , 1 ], define a probability measure β„™ΞΌ,nsubscriptβ„™πœ‡π‘›\mathbb{P}_{\mu,n}roman_β„™ start_POSTSUBSCRIPT italic_ΞΌ , italic_n end_POSTSUBSCRIPT on {0,1}nsuperscript01𝑛\{0,1\}^{n}{ 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT as

(2.2.1) β„™ΞΌ,n⁒(A)β‰”βˆ«01Ξ½p⁒(A)⁒μ⁒(d⁒p),AβŠ‚{0,1}n,formulae-sequence≔subscriptβ„™πœ‡π‘›π΄superscriptsubscript01subscriptπœˆπ‘π΄πœ‡π‘‘π‘π΄superscript01𝑛\mathbb{P}_{\mu,n}(A)\coloneqq\int_{0}^{1}\nu_{p}(A)\mu(dp),\quad A\subset\{0,% 1\}^{n},roman_β„™ start_POSTSUBSCRIPT italic_ΞΌ , italic_n end_POSTSUBSCRIPT ( italic_A ) ≔ ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT italic_Ξ½ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_A ) italic_ΞΌ ( italic_d italic_p ) , italic_A βŠ‚ { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ,

where Ξ½psubscriptπœˆπ‘\nu_{p}italic_Ξ½ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT denotes the Bernoulli measure on {0,1}n.superscript01𝑛\{0,1\}^{n}.{ 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT . Recall that the map μ↦ℙμ,nmaps-toπœ‡subscriptβ„™πœ‡π‘›\mu\mapsto{\mathbb{P}}_{\mu,n}italic_ΞΌ ↦ roman_β„™ start_POSTSUBSCRIPT italic_ΞΌ , italic_n end_POSTSUBSCRIPT is not surjective.

Let Ο€ksubscriptπœ‹π‘˜\pi_{k}italic_Ο€ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT denote the canonical projection from {0,1}nsuperscript01𝑛\{0,1\}^{n}{ 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT onto its first kπ‘˜kitalic_k coordinates, and let β„™ksubscriptβ„™π‘˜\mathbb{P}_{k}roman_β„™ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT denote the pushforward of β„™β„™\mathbb{P}roman_β„™ under Ο€ksubscriptπœ‹π‘˜\pi_{k}italic_Ο€ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. Clearly, (β„™ΞΌ,n)k=β„™ΞΌ,ksubscriptsubscriptβ„™πœ‡π‘›π‘˜subscriptβ„™πœ‡π‘˜(\mathbb{P}_{\mu,n})_{k}=\mathbb{P}_{\mu,k}( roman_β„™ start_POSTSUBSCRIPT italic_ΞΌ , italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = roman_β„™ start_POSTSUBSCRIPT italic_ΞΌ , italic_k end_POSTSUBSCRIPT. The variational distance between two probability measures ΞΌπœ‡\muitalic_ΞΌ and ν𝜈\nuitalic_Ξ½ on (Ξ©,β„±)Ξ©β„±(\Omega,\mathcal{F})( roman_Ξ© , caligraphic_F ) is defined as

β€–ΞΌβˆ’Ξ½β€–β‰”2⁒supAβˆˆβ„±|μ⁒(A)βˆ’Ξ½β’(A)|.≔normπœ‡πœˆ2subscriptsupremumπ΄β„±πœ‡π΄πœˆπ΄\|\mu-\nu\|\coloneqq 2\,\sup\limits_{A\in\mathcal{F}}|\mu(A)-\nu(A)|.βˆ₯ italic_ΞΌ - italic_Ξ½ βˆ₯ ≔ 2 roman_sup start_POSTSUBSCRIPT italic_A ∈ caligraphic_F end_POSTSUBSCRIPT | italic_ΞΌ ( italic_A ) - italic_Ξ½ ( italic_A ) | .
Theorem 2.2.2.

(Diaconis-Freedman) Let β„™β„™\mathbb{P}roman_β„™ be an exchangeable measure on {0,1}nsuperscript01𝑛\{0,1\}^{n}{ 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. Then there exists a probability measure ΞΌπœ‡\muitalic_ΞΌ on [0,1]01[0,1][ 0 , 1 ] such that

(2.2.3) β€–β„™kβˆ’β„™ΞΌ,kβ€–β©½4⁒kn,for all β’kβ©½n,formulae-sequencenormsubscriptβ„™π‘˜subscriptβ„™πœ‡π‘˜4π‘˜π‘›for all π‘˜π‘›\|\mathbb{P}_{k}-\mathbb{P}_{\mu,k}\|\leqslant\frac{4k}{n},\quad\text{for all % }k\leqslant n,βˆ₯ roman_β„™ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - roman_β„™ start_POSTSUBSCRIPT italic_ΞΌ , italic_k end_POSTSUBSCRIPT βˆ₯ β©½ divide start_ARG 4 italic_k end_ARG start_ARG italic_n end_ARG , for all italic_k β©½ italic_n ,

and this rate of convergence is sharp.

2.3. Extreme measures

The spaces of exchangeable and qπ‘žqitalic_q-exchangeable probability measures are convex and compact; hence, by Choquet’s theorem, they are the closed convex hulls of their extreme points.

Proposition 2.3.1.
  1. (1)

    Let Ξ©={0,1}∞Ωsuperscript01\Omega=\{0,1\}^{\infty}roman_Ξ© = { 0 , 1 } start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT. The extreme points of the set of exchangeable probability measures on ΩΩ\Omegaroman_Ξ© are precisely the Bernoulli measures Ξ½psubscriptπœˆπ‘\nu_{p}italic_Ξ½ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT, with p∈[0,1]𝑝01p\in[0,1]italic_p ∈ [ 0 , 1 ]. For qπ‘žqitalic_q-exchangeable measures, the extreme ones are parametrized by xβˆˆΞ”qπ‘₯subscriptΞ”π‘žx\in\Delta_{q}italic_x ∈ roman_Ξ” start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT and are given by the measures Ξ½xqsubscriptsuperscriptπœˆπ‘žπ‘₯\nu^{q}_{x}italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT defined in  (2.1.5).

  2. (2)

    Let Ξ©={0,1}nΞ©superscript01𝑛\Omega=\{0,1\}^{n}roman_Ξ© = { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. In this case, the sets of extreme exchangeable and qπ‘žqitalic_q-exchangeable measures are finite. The extreme qπ‘žqitalic_q-exchangeable measures, denoted by e0q,e1q,…,enqsubscriptsuperscriptπ‘’π‘ž0subscriptsuperscriptπ‘’π‘ž1…subscriptsuperscriptπ‘’π‘žπ‘›e^{q}_{0},e^{q}_{1},\ldots,e^{q}_{n}italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, are given by the formula

    ekq⁒(Ο‰)subscriptsuperscriptπ‘’π‘žπ‘˜πœ”\displaystyle e^{q}_{k}(\omega)italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_Ο‰ ) ={qinv⁒(Ο‰)⁒1[nk],if β’ω⁒ contains β’k⁒ ones and β’nβˆ’k⁒ zeros,0,otherwise.absentcasessuperscriptπ‘žinvπœ”1FRACOPπ‘›π‘˜if πœ” contains π‘˜ ones and π‘›π‘˜ zeros0otherwise.\displaystyle=\begin{cases}q^{\text{inv}(\omega)}\dfrac{1}{\genfrac{[}{]}{0.0% pt}{}{n}{k}},&\text{if }\omega\text{ contains }k\text{ ones and }n-k\text{ % zeros},\\ 0,&\text{otherwise.}\end{cases}= { start_ROW start_CELL italic_q start_POSTSUPERSCRIPT inv ( italic_Ο‰ ) end_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG [ FRACOP start_ARG italic_n end_ARG start_ARG italic_k end_ARG ] end_ARG , end_CELL start_CELL if italic_Ο‰ contains italic_k ones and italic_n - italic_k zeros , end_CELL end_ROW start_ROW start_CELL 0 , end_CELL start_CELL otherwise. end_CELL end_ROW

    Setting q=1π‘ž1q=1italic_q = 1, we obtain the extreme measures in the classical exchangeable case.

Proof.

Claim (1) follows immediately from the bijections in (1.0.1) and (2.1.7). For Claim (2), note that due to qπ‘žqitalic_q-exchangeability, the probability depends only on the number of ones, up to the scalar factor qinv⁑(Ο‰)superscriptπ‘žinvπœ”q^{\operatorname{inv}(\omega)}italic_q start_POSTSUPERSCRIPT roman_inv ( italic_Ο‰ ) end_POSTSUPERSCRIPT. This shows that each ekqsubscriptsuperscriptπ‘’π‘žπ‘˜e^{q}_{k}italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is extreme. ∎

Fix n1β©½nsubscript𝑛1𝑛n_{1}\leqslant nitalic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT β©½ italic_n. For the extreme measure en,n1qsubscriptsuperscriptπ‘’π‘žπ‘›subscript𝑛1e^{q}_{n,n_{1}}italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT and the measure Ξ½xqsubscriptsuperscriptπœˆπ‘žπ‘₯\nu^{q}_{x}italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT with parameter x=qn1π‘₯superscriptπ‘žsubscript𝑛1x=q^{n_{1}}italic_x = italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT, we compute the probabilities in (2.3.3) and (2.3.4), corresponding to the event that the first kπ‘˜kitalic_k entries of the sequence begin with exactly k1subscriptπ‘˜1k_{1}italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ones.

Proposition 2.3.2.
(2.3.3) (en,n1q)k⁒(sk,k1)=q(n1βˆ’k1)⁒(kβˆ’k1)⁒[nβˆ’kn1βˆ’k1]/[nn1],subscriptsubscriptsuperscriptπ‘’π‘žπ‘›subscript𝑛1π‘˜subscriptsπ‘˜subscriptπ‘˜1superscriptπ‘žsubscript𝑛1subscriptπ‘˜1π‘˜subscriptπ‘˜1FRACOPπ‘›π‘˜subscript𝑛1subscriptπ‘˜1FRACOP𝑛subscript𝑛1(e^{q}_{n,n_{1}})_{k}\left(\text{s}_{k,k_{1}}\right)=q^{(n_{1}-k_{1})(k-k_{1})% }\genfrac{[}{]}{0.0pt}{}{n-k}{n_{1}-k_{1}}\bigg{/}\genfrac{[}{]}{0.0pt}{}{n}{n% _{1}},( italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( s start_POSTSUBSCRIPT italic_k , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) = italic_q start_POSTSUPERSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT [ FRACOP start_ARG italic_n - italic_k end_ARG start_ARG italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ] / [ FRACOP start_ARG italic_n end_ARG start_ARG italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ] ,
(2.3.4) (Ξ½xq)k⁒(sk,k1)=q(n1βˆ’k1)⁒(kβˆ’k1)⁒(qn1;qβˆ’1)k1.subscriptsubscriptsuperscriptπœˆπ‘žπ‘₯π‘˜subscriptsπ‘˜subscriptπ‘˜1superscriptπ‘žsubscript𝑛1subscriptπ‘˜1π‘˜subscriptπ‘˜1subscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1subscriptπ‘˜1(\nu^{q}_{x})_{k}\left(\text{s}_{k,k_{1}}\right)=q^{(n_{1}-k_{1})(k-k_{1})}(q^% {n_{1}};q^{-1})_{k_{1}}.( italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( s start_POSTSUBSCRIPT italic_k , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) = italic_q start_POSTSUPERSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT .
Proof.

We prove only (2.3.3), the computation for (2.3.4) is analogous. We have

(en,n1q)k⁒(sk,k1)=βˆ‘Ο‰~∈{0,1}nβˆ’ken,n1q⁒(sk,k1βˆͺΟ‰~),subscriptsubscriptsuperscriptπ‘’π‘žπ‘›subscript𝑛1π‘˜subscriptsπ‘˜subscriptπ‘˜1subscript~πœ”superscript01π‘›π‘˜subscriptsuperscriptπ‘’π‘žπ‘›subscript𝑛1subscriptsπ‘˜subscriptπ‘˜1~πœ”(e^{q}_{n,n_{1}})_{k}\left(\text{s}_{k,k_{1}}\right)=\sum_{\tilde{\omega}\in\{% 0,1\}^{n-k}}e^{q}_{n,n_{1}}\left(\text{s}_{k,k_{1}}\cup\tilde{\omega}\right),( italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( s start_POSTSUBSCRIPT italic_k , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) = βˆ‘ start_POSTSUBSCRIPT over~ start_ARG italic_Ο‰ end_ARG ∈ { 0 , 1 } start_POSTSUPERSCRIPT italic_n - italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( s start_POSTSUBSCRIPT italic_k , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT βˆͺ over~ start_ARG italic_Ο‰ end_ARG ) ,

where sk,k1βˆͺΟ‰~subscriptsπ‘˜subscriptπ‘˜1~πœ”\text{s}_{k,k_{1}}\cup\tilde{\omega}s start_POSTSUBSCRIPT italic_k , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT βˆͺ over~ start_ARG italic_Ο‰ end_ARG denotes the concatenation of two sequences. By counting inversions, we obtain

βˆ‘Ο‰~∈{0,1}nβˆ’ken,n1q⁒(1,…,1,0,…,0,Ο‰~)=[nβˆ’kn1βˆ’k1]⁒en,n1q⁒(sk,k1βˆͺsnβˆ’k,n1βˆ’k1),subscript~πœ”superscript01π‘›π‘˜subscriptsuperscriptπ‘’π‘žπ‘›subscript𝑛11…10…0~πœ”FRACOPπ‘›π‘˜subscript𝑛1subscriptπ‘˜1subscriptsuperscriptπ‘’π‘žπ‘›subscript𝑛1subscriptsπ‘˜subscriptπ‘˜1subscriptsπ‘›π‘˜subscript𝑛1subscriptπ‘˜1\sum_{\tilde{\omega}\in\{0,1\}^{n-k}}e^{q}_{n,n_{1}}(1,\ldots,1,0,\ldots,0,% \tilde{\omega})=\genfrac{[}{]}{0.0pt}{}{n-k}{n_{1}-k_{1}}e^{q}_{n,n_{1}}\left(% \text{s}_{k,k_{1}}\cup\text{s}_{n-k,n_{1}-k_{1}}\right),βˆ‘ start_POSTSUBSCRIPT over~ start_ARG italic_Ο‰ end_ARG ∈ { 0 , 1 } start_POSTSUPERSCRIPT italic_n - italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( 1 , … , 1 , 0 , … , 0 , over~ start_ARG italic_Ο‰ end_ARG ) = [ FRACOP start_ARG italic_n - italic_k end_ARG start_ARG italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ] italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( s start_POSTSUBSCRIPT italic_k , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT βˆͺ s start_POSTSUBSCRIPT italic_n - italic_k , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ,

where sk,k1βˆͺsnβˆ’k,n1βˆ’k1subscriptsπ‘˜subscriptπ‘˜1subscriptsπ‘›π‘˜subscript𝑛1subscriptπ‘˜1\text{s}_{k,k_{1}}\cup\text{s}_{n-k,n_{1}-k_{1}}s start_POSTSUBSCRIPT italic_k , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT βˆͺ s start_POSTSUBSCRIPT italic_n - italic_k , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT is the concatenation of two sequences of the same form. The number of inversions in sk,k1βˆͺsnβˆ’k,n1βˆ’k1subscriptsπ‘˜subscriptπ‘˜1subscriptsπ‘›π‘˜subscript𝑛1subscriptπ‘˜1\text{s}_{k,k_{1}}\cup\text{s}_{n-k,n_{1}-k_{1}}s start_POSTSUBSCRIPT italic_k , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT βˆͺ s start_POSTSUBSCRIPT italic_n - italic_k , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT equals (n1βˆ’k1)⁒(kβˆ’k1)subscript𝑛1subscriptπ‘˜1π‘˜subscriptπ‘˜1(n_{1}-k_{1})(k-k_{1})( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ). Therefore,

[nβˆ’kn1βˆ’k1]⁒en,n1q⁒(sk,k1βˆͺsnβˆ’k,n1βˆ’k1)FRACOPπ‘›π‘˜subscript𝑛1subscriptπ‘˜1subscriptsuperscriptπ‘’π‘žπ‘›subscript𝑛1subscriptsπ‘˜subscriptπ‘˜1subscriptsπ‘›π‘˜subscript𝑛1subscriptπ‘˜1\displaystyle\genfrac{[}{]}{0.0pt}{}{n-k}{n_{1}-k_{1}}e^{q}_{n,n_{1}}\left(% \text{s}_{k,k_{1}}\cup\text{s}_{n-k,n_{1}-k_{1}}\right)[ FRACOP start_ARG italic_n - italic_k end_ARG start_ARG italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ] italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( s start_POSTSUBSCRIPT italic_k , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT βˆͺ s start_POSTSUBSCRIPT italic_n - italic_k , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) =q(n1βˆ’k1)⁒(kβˆ’k1)⁒[nβˆ’kn1βˆ’k1]⁒en,n1q⁒(sn,n1)absentsuperscriptπ‘žsubscript𝑛1subscriptπ‘˜1π‘˜subscriptπ‘˜1FRACOPπ‘›π‘˜subscript𝑛1subscriptπ‘˜1subscriptsuperscriptπ‘’π‘žπ‘›subscript𝑛1subscripts𝑛subscript𝑛1\displaystyle=q^{(n_{1}-k_{1})(k-k_{1})}\genfrac{[}{]}{0.0pt}{}{n-k}{n_{1}-k_{% 1}}e^{q}_{n,n_{1}}\left(\text{s}_{n,n_{1}}\right)= italic_q start_POSTSUPERSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT [ FRACOP start_ARG italic_n - italic_k end_ARG start_ARG italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ] italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( s start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT )
=q(n1βˆ’k1)⁒(kβˆ’k1)⁒[nβˆ’kn1βˆ’k1]/[nn1],absentsuperscriptπ‘žsubscript𝑛1subscriptπ‘˜1π‘˜subscriptπ‘˜1FRACOPπ‘›π‘˜subscript𝑛1subscriptπ‘˜1FRACOP𝑛subscript𝑛1\displaystyle=q^{(n_{1}-k_{1})(k-k_{1})}\genfrac{[}{]}{0.0pt}{}{n-k}{n_{1}-k_{% 1}}\bigg{/}\genfrac{[}{]}{0.0pt}{}{n}{n_{1}},= italic_q start_POSTSUPERSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT [ FRACOP start_ARG italic_n - italic_k end_ARG start_ARG italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ] / [ FRACOP start_ARG italic_n end_ARG start_ARG italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ] ,

which proves the claim. ∎

3. Finite Form

3.1. Main result

In this section, we formulate an asymptotic version of Theorem (2.1.6) in the sense of Theorem  (2.2.2). Abusing notation, for a probability measure ΞΌπœ‡\muitalic_ΞΌ on Ξ”qsubscriptΞ”π‘ž\Delta_{q}roman_Ξ” start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT, we denote by β„™ΞΌ,nsubscriptβ„™πœ‡π‘›{\mathbb{P}}_{\mu,n}roman_β„™ start_POSTSUBSCRIPT italic_ΞΌ , italic_n end_POSTSUBSCRIPT the probability measure given by

β„™ΞΌ,n⁒(A)=βˆ«Ξ”qΞ½xq⁒(A)⁒μ⁒(d⁒x),AβŠ‚{0,1}n,formulae-sequencesubscriptβ„™πœ‡π‘›π΄subscriptsubscriptΞ”π‘žsubscriptsuperscriptπœˆπ‘žπ‘₯π΄πœ‡π‘‘π‘₯𝐴superscript01𝑛{\mathbb{P}}_{\mu,n}(A)=\int_{\Delta_{q}}\nu^{q}_{x}(A)\,\mu(dx),\quad A% \subset\{0,1\}^{n},roman_β„™ start_POSTSUBSCRIPT italic_ΞΌ , italic_n end_POSTSUBSCRIPT ( italic_A ) = ∫ start_POSTSUBSCRIPT roman_Ξ” start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_A ) italic_ΞΌ ( italic_d italic_x ) , italic_A βŠ‚ { 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ,

where Ξ½xqsubscriptsuperscriptπœˆπ‘žπ‘₯\nu^{q}_{x}italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT denotes a probability measure on {0,1}nsuperscript01𝑛\{0,1\}^{n}{ 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT defined by (2.1.5).

Theorem 3.1.1.

Let β„™β„™\mathbb{P}roman_β„™ be an qπ‘žqitalic_q-exchangeable probability measure on {0,1}nsuperscript01𝑛\{0,1\}^{n}{ 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. Then there exists a probability measure ΞΌπœ‡\muitalic_ΞΌ on Ξ”qsubscriptΞ”π‘ž\Delta_{q}roman_Ξ” start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT such that

(3.1.2) β€–β„™kβˆ’β„™ΞΌ,kβ€–β©½ckβ‹…qn,for all β’kβ©½n,formulae-sequencenormsubscriptβ„™π‘˜subscriptβ„™πœ‡π‘˜β‹…subscriptπ‘π‘˜superscriptπ‘žπ‘›for all π‘˜π‘›\|\mathbb{P}_{k}-\mathbb{P}_{\mu,k}\|\leqslant c_{k}\cdot q^{n},\quad\text{for% all }k\leqslant n,βˆ₯ roman_β„™ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - roman_β„™ start_POSTSUBSCRIPT italic_ΞΌ , italic_k end_POSTSUBSCRIPT βˆ₯ β©½ italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT β‹… italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , for all italic_k β©½ italic_n ,

where cksubscriptπ‘π‘˜c_{k}italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is a constant depending only on kπ‘˜kitalic_k.

The convergence rate of order qnsuperscriptπ‘žπ‘›q^{n}italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT is sharp, as will be shown in Section 3.4. For convenience, we do not write the constant explicitly, only its existence is relevant for our purposes. Using the convex structure of the space of qπ‘žqitalic_q-exchangeable measures, we reduce the proof of Theorem  (3.1.1) to the case of an extreme measure.

Lemma 3.1.3.

Fix n1∈{0,1,…,n}subscript𝑛101…𝑛n_{1}\in\{0,1,\ldots,n\}italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ { 0 , 1 , … , italic_n }. In the notation of Theorem (3.1.1), consider the extreme measure β„™=en,n1qβ„™subscriptsuperscriptπ‘’π‘žπ‘›subscript𝑛1\mathbb{P}=e^{q}_{n,n_{1}}roman_β„™ = italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT and the probability measure ΞΌ=Ξ΄qn1πœ‡subscript𝛿superscriptπ‘žsubscript𝑛1\mu=\delta_{q^{n_{1}}}italic_ΞΌ = italic_Ξ΄ start_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT. Then

(3.1.4) β€–β„™kβˆ’β„™ΞΌ,kβ€–=β€–(en,n1q)kβˆ’(Ξ½qn1q)kβ€–β©½ckβ‹…qn,for all β’kβ©½n,formulae-sequencenormsubscriptβ„™π‘˜subscriptβ„™πœ‡π‘˜normsubscriptsubscriptsuperscriptπ‘’π‘žπ‘›subscript𝑛1π‘˜subscriptsubscriptsuperscriptπœˆπ‘žsuperscriptπ‘žsubscript𝑛1π‘˜β‹…subscriptπ‘π‘˜superscriptπ‘žπ‘›for all π‘˜π‘›\|{\mathbb{P}}_{k}-{\mathbb{P}}_{\mu,k}\|=\|(e^{q}_{n,n_{1}})_{k}-(\nu^{q}_{q^% {n_{1}}})_{k}\|\leqslant c_{k}\cdot q^{n},\quad\text{for all }k\leqslant n,βˆ₯ roman_β„™ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - roman_β„™ start_POSTSUBSCRIPT italic_ΞΌ , italic_k end_POSTSUBSCRIPT βˆ₯ = βˆ₯ ( italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - ( italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT βˆ₯ β©½ italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT β‹… italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , for all italic_k β©½ italic_n ,

where cksubscriptπ‘π‘˜c_{k}italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is a constant depending only on kπ‘˜kitalic_k.

Note that the estimate (3.1.4) is uniform in the parameter n1subscript𝑛1n_{1}italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. The proof of the lemma is given in section  (3.2). We now apply this lemma to prove the theorem.

Proof of Theorem  (3.1.1).

Consider a convex decomposition of the measure β„™β„™{\mathbb{P}}roman_β„™

β„™=Ξ±0⁒e0q+…+Ξ±n⁒enq,βˆ‘i=0nΞ±i=1.formulae-sequenceβ„™subscript𝛼0subscriptsuperscriptπ‘’π‘ž0…subscript𝛼𝑛subscriptsuperscriptπ‘’π‘žπ‘›superscriptsubscript𝑖0𝑛subscript𝛼𝑖1{\mathbb{P}}=\alpha_{0}e^{q}_{0}+\ldots+\alpha_{n}e^{q}_{n},\quad\sum_{i=0}^{n% }\alpha_{i}=1.roman_β„™ = italic_Ξ± start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + … + italic_Ξ± start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , βˆ‘ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_Ξ± start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1 .

Then the corresponding pushforward measure is given by

β„™k=Ξ±0⁒(e0q)k+…+Ξ±n⁒(enq)k.subscriptβ„™π‘˜subscript𝛼0subscriptsubscriptsuperscriptπ‘’π‘ž0π‘˜β€¦subscript𝛼𝑛subscriptsubscriptsuperscriptπ‘’π‘žπ‘›π‘˜{\mathbb{P}}_{k}=\alpha_{0}(e^{q}_{0})_{k}+\ldots+\alpha_{n}(e^{q}_{n})_{k}.roman_β„™ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_Ξ± start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + … + italic_Ξ± start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT .

Define the probability measure ΞΌπœ‡\muitalic_ΞΌ by setting μ⁒(qi):=Ξ±i.assignπœ‡superscriptπ‘žπ‘–subscript𝛼𝑖\mu(q^{i}):=\alpha_{i}.italic_ΞΌ ( italic_q start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) := italic_Ξ± start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT . Now consider the variation distance

β€–β„™kβˆ’β„™ΞΌ,kβ€–=β€–βˆ‘i=0nΞ±i⁒(en,iq)kβˆ’βˆ‘i=0nΞ±i⁒(Ξ½xq)kβ€–β©½βˆ‘i=0nΞ±i⁒‖(en,iq)kβˆ’(Ξ½xq)kβ€–.normsubscriptβ„™π‘˜subscriptβ„™πœ‡π‘˜normsuperscriptsubscript𝑖0𝑛subscript𝛼𝑖subscriptsubscriptsuperscriptπ‘’π‘žπ‘›π‘–π‘˜superscriptsubscript𝑖0𝑛subscript𝛼𝑖subscriptsubscriptsuperscriptπœˆπ‘žπ‘₯π‘˜superscriptsubscript𝑖0𝑛subscript𝛼𝑖normsubscriptsubscriptsuperscriptπ‘’π‘žπ‘›π‘–π‘˜subscriptsubscriptsuperscriptπœˆπ‘žπ‘₯π‘˜\|{\mathbb{P}}_{k}-{\mathbb{P}}_{\mu,k}\|=\|\sum_{i=0}^{n}\alpha_{i}(e^{q}_{n,% i})_{k}-\sum_{i=0}^{n}\alpha_{i}(\nu^{q}_{x})_{k}\|\leqslant\sum_{i=0}^{n}% \alpha_{i}\|(e^{q}_{n,i})_{k}-(\nu^{q}_{x})_{k}\|.βˆ₯ roman_β„™ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - roman_β„™ start_POSTSUBSCRIPT italic_ΞΌ , italic_k end_POSTSUBSCRIPT βˆ₯ = βˆ₯ βˆ‘ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_Ξ± start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_i end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - βˆ‘ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_Ξ± start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT βˆ₯ β©½ βˆ‘ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_Ξ± start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT βˆ₯ ( italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_i end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - ( italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT βˆ₯ .

Finally, applying Lemma  (3.1.3), we obtain β€–β„™kβˆ’β„™ΞΌ,kβ€–β©½ckβ‹…qnnormsubscriptβ„™π‘˜subscriptβ„™πœ‡π‘˜β‹…subscriptπ‘π‘˜superscriptπ‘žπ‘›\|{\mathbb{P}}_{k}-{\mathbb{P}}_{\mu,k}\|\leqslant c_{k}\cdot q^{n}βˆ₯ roman_β„™ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - roman_β„™ start_POSTSUBSCRIPT italic_ΞΌ , italic_k end_POSTSUBSCRIPT βˆ₯ β©½ italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT β‹… italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. ∎

3.2. Extreme case

In this section, we prove Lemma (3.1.3).

Proof.

The variational distance (3.1.4) between the corresponding pushforward measures can be computed as follows

β€–(en,n1q)kβˆ’(Ξ½qn1q)kβ€–normsubscriptsubscriptsuperscriptπ‘’π‘žπ‘›subscript𝑛1π‘˜subscriptsubscriptsuperscriptπœˆπ‘žsuperscriptπ‘žsubscript𝑛1π‘˜\displaystyle\big{\|}(e^{q}_{n,n_{1}})_{k}-(\nu^{q}_{q^{n_{1}}})_{k}\big{\|}βˆ₯ ( italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - ( italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT βˆ₯ =βˆ‘Ο‰βˆˆ{0,1}k|(en,n1q)k⁒(Ο‰)βˆ’(Ξ½qn1q)k⁒(Ο‰)|absentsubscriptπœ”superscript01π‘˜subscriptsubscriptsuperscriptπ‘’π‘žπ‘›subscript𝑛1π‘˜πœ”subscriptsubscriptsuperscriptπœˆπ‘žsuperscriptπ‘žsubscript𝑛1π‘˜πœ”\displaystyle=\sum_{\omega\in\{0,1\}^{k}}\left|(e^{q}_{n,n_{1}})_{k}(\omega)-(% \nu^{q}_{q^{n_{1}}})_{k}(\omega)\right|= βˆ‘ start_POSTSUBSCRIPT italic_Ο‰ ∈ { 0 , 1 } start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | ( italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_Ο‰ ) - ( italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_Ο‰ ) |
=βˆ‘k1=0kβˆ‘Ο‰βˆˆCk,k1qinv⁒(Ο‰)⁒|(en,n1q)k⁒(sk,k1)βˆ’(Ξ½qn1q)k⁒(sk,k1)|absentsuperscriptsubscriptsubscriptπ‘˜10π‘˜subscriptπœ”subscriptπΆπ‘˜subscriptπ‘˜1superscriptπ‘žinvπœ”subscriptsubscriptsuperscriptπ‘’π‘žπ‘›subscript𝑛1π‘˜subscriptsπ‘˜subscriptπ‘˜1subscriptsubscriptsuperscriptπœˆπ‘žsuperscriptπ‘žsubscript𝑛1π‘˜subscriptsπ‘˜subscriptπ‘˜1\displaystyle=\sum_{k_{1}=0}^{k}\sum_{\omega\in C_{k,k_{1}}}q^{\text{inv}(% \omega)}\left|(e^{q}_{n,n_{1}})_{k}(\text{s}_{k,k_{1}})-(\nu^{q}_{q^{n_{1}}})_% {k}(\text{s}_{k,k_{1}})\right|= βˆ‘ start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT βˆ‘ start_POSTSUBSCRIPT italic_Ο‰ ∈ italic_C start_POSTSUBSCRIPT italic_k , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT inv ( italic_Ο‰ ) end_POSTSUPERSCRIPT | ( italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( s start_POSTSUBSCRIPT italic_k , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) - ( italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( s start_POSTSUBSCRIPT italic_k , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) |
=βˆ‘k1=0k[kk1]⁒|(en,n1q)k⁒(sk,k1)βˆ’(Ξ½qn1q)k⁒(sk,k1)|absentsuperscriptsubscriptsubscriptπ‘˜10π‘˜FRACOPπ‘˜subscriptπ‘˜1subscriptsubscriptsuperscriptπ‘’π‘žπ‘›subscript𝑛1π‘˜subscriptsπ‘˜subscriptπ‘˜1subscriptsubscriptsuperscriptπœˆπ‘žsuperscriptπ‘žsubscript𝑛1π‘˜subscriptsπ‘˜subscriptπ‘˜1\displaystyle=\sum_{k_{1}=0}^{k}\genfrac{[}{]}{0.0pt}{}{k}{k_{1}}\left|(e^{q}_% {n,n_{1}})_{k}(\text{s}_{k,k_{1}})-(\nu^{q}_{q^{n_{1}}})_{k}(\text{s}_{k,k_{1}% })\right|= βˆ‘ start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT [ FRACOP start_ARG italic_k end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ] | ( italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( s start_POSTSUBSCRIPT italic_k , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) - ( italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( s start_POSTSUBSCRIPT italic_k , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) |
=βˆ‘k1=0k[kk1]⁒q(n1βˆ’k1)⁒(kβˆ’k1)⁒|[nβˆ’kn1βˆ’k1]/[nn1]βˆ’(qn1;qβˆ’1)k1|.absentsuperscriptsubscriptsubscriptπ‘˜10π‘˜FRACOPπ‘˜subscriptπ‘˜1superscriptπ‘žsubscript𝑛1subscriptπ‘˜1π‘˜subscriptπ‘˜1FRACOPπ‘›π‘˜subscript𝑛1subscriptπ‘˜1FRACOP𝑛subscript𝑛1subscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1subscriptπ‘˜1\displaystyle=\sum_{k_{1}=0}^{k}\genfrac{[}{]}{0.0pt}{}{k}{k_{1}}\,q^{(n_{1}-k% _{1})(k-k_{1})}\left|\genfrac{[}{]}{0.0pt}{}{n-k}{n_{1}-k_{1}}\bigg{/}\genfrac% {[}{]}{0.0pt}{}{n}{n_{1}}-(q^{n_{1}};q^{-1})_{k_{1}}\right|.= βˆ‘ start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT [ FRACOP start_ARG italic_k end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ] italic_q start_POSTSUPERSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT | [ FRACOP start_ARG italic_n - italic_k end_ARG start_ARG italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ] / [ FRACOP start_ARG italic_n end_ARG start_ARG italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ] - ( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | .

where the second identity follows from the qπ‘žqitalic_q-exchangeability property, the third from Proposition (2.1.4), and the fourth from Proposition (2.3.3). It follows that it suffices to analyse the expression

(3.2.1) q(n1βˆ’k1)⁒(kβˆ’k1)⁒|[nβˆ’kn1βˆ’k1]/[nn1]βˆ’(qn1;qβˆ’1)k1|=superscriptπ‘žsubscript𝑛1subscriptπ‘˜1π‘˜subscriptπ‘˜1FRACOPπ‘›π‘˜subscript𝑛1subscriptπ‘˜1FRACOP𝑛subscript𝑛1subscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1subscriptπ‘˜1absentq^{(n_{1}-k_{1})(k-k_{1})}\left|\genfrac{[}{]}{0.0pt}{}{n-k}{n_{1}-k_{1}}\bigg% {/}\genfrac{[}{]}{0.0pt}{}{n}{n_{1}}-(q^{n_{1}};q^{-1})_{k_{1}}\right|=italic_q start_POSTSUPERSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT | [ FRACOP start_ARG italic_n - italic_k end_ARG start_ARG italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ] / [ FRACOP start_ARG italic_n end_ARG start_ARG italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ] - ( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | =
q(n1βˆ’k1)⁒(kβˆ’k1)⁒|[nβˆ’k]![n]!⁒[n1]![n1βˆ’k1]!⁒[nβˆ’n1]![nβˆ’n1βˆ’(kβˆ’k1)]!βˆ’(qn1;qβˆ’1)k1|.superscriptπ‘žsubscript𝑛1subscriptπ‘˜1π‘˜subscriptπ‘˜1delimited-[]π‘›π‘˜delimited-[]𝑛delimited-[]subscript𝑛1delimited-[]subscript𝑛1subscriptπ‘˜1delimited-[]𝑛subscript𝑛1delimited-[]𝑛subscript𝑛1π‘˜subscriptπ‘˜1subscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1subscriptπ‘˜1q^{(n_{1}-k_{1})(k-k_{1})}\left|\dfrac{[n-k]!}{[n]!}\dfrac{[n_{1}]!}{[n_{1}-k_% {1}]!}\dfrac{[n-n_{1}]!}{[n-n_{1}-(k-k_{1})]!}-(q^{n_{1}};q^{-1})_{k_{1}}% \right|.italic_q start_POSTSUPERSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT | divide start_ARG [ italic_n - italic_k ] ! end_ARG start_ARG [ italic_n ] ! end_ARG divide start_ARG [ italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ] ! end_ARG start_ARG [ italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ] ! end_ARG divide start_ARG [ italic_n - italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ] ! end_ARG start_ARG [ italic_n - italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - ( italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ] ! end_ARG - ( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | .

We consider two cases: k1=ksubscriptπ‘˜1π‘˜k_{1}=kitalic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_k and k1<ksubscriptπ‘˜1π‘˜k_{1}<kitalic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_k.

Case 1: k1=ksubscriptπ‘˜1π‘˜k_{1}=kitalic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_k. In this case, (3.2.1) reduces to

(3.2.2) |[nβˆ’k]![n]!⁒[n1]![n1βˆ’k]!βˆ’(qn1;qβˆ’1)k|delimited-[]π‘›π‘˜delimited-[]𝑛delimited-[]subscript𝑛1delimited-[]subscript𝑛1π‘˜subscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1π‘˜\displaystyle\left|\dfrac{[n-k]!}{[n]!}\dfrac{[n_{1}]!}{[n_{1}-k]!}\ -(q^{n_{1% }};q^{-1})_{k}\right|| divide start_ARG [ italic_n - italic_k ] ! end_ARG start_ARG [ italic_n ] ! end_ARG divide start_ARG [ italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ] ! end_ARG start_ARG [ italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k ] ! end_ARG - ( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | =|(qn1;qβˆ’1)k∏i=0kβˆ’1(1βˆ’qnβˆ’i)βˆ’(qn1;qβˆ’1)k|absentsubscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1π‘˜superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–subscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1π‘˜\displaystyle=\left|\dfrac{(q^{n_{1}};q^{-1})_{k}}{\prod_{i=0}^{k-1}(1-q^{n-i}% )}-(q^{n_{1}};q^{-1})_{k}\right|= | divide start_ARG ( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG - ( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |
=(qn1;qβˆ’1)k⁒1βˆ’βˆi=0kβˆ’1(1βˆ’qnβˆ’i)∏i=0kβˆ’1(1βˆ’qnβˆ’i),absentsubscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1π‘˜1superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–\displaystyle=(q^{n_{1}};q^{-1})_{k}\dfrac{1-\prod_{i=0}^{k-1}(1-q^{n-i})}{% \prod_{i=0}^{k-1}(1-q^{n-i})},= ( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT divide start_ARG 1 - ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG start_ARG ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG ,

since (qn1;qβˆ’1)kβ©½1subscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1π‘˜1(q^{n_{1}};q^{-1})_{k}\leqslant 1( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT β©½ 1 and ∏i=0kβˆ’1(1βˆ’qnβˆ’i)β©Ύ(1βˆ’q)ksuperscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–superscript1π‘žπ‘˜{\prod_{i=0}^{k-1}(1-q^{n-i})}\geqslant(1-q)^{k}∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) β©Ύ ( 1 - italic_q ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT, we obtain the estimate

(3.2.3) (qn1;qβˆ’1)k⁒1βˆ’βˆi=0kβˆ’1(1βˆ’qnβˆ’i)∏i=0kβˆ’1(1βˆ’qnβˆ’i)β©½1βˆ’βˆi=0kβˆ’1(1βˆ’qnβˆ’i)(1βˆ’q)k,subscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1π‘˜1superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–1superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–superscript1π‘žπ‘˜(q^{n_{1}};q^{-1})_{k}\dfrac{1-\prod_{i=0}^{k-1}(1-q^{n-i})}{\prod_{i=0}^{k-1}% (1-q^{n-i})}\leqslant\dfrac{1-\prod_{i=0}^{k-1}(1-q^{n-i})}{(1-q)^{k}},( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT divide start_ARG 1 - ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG start_ARG ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG β©½ divide start_ARG 1 - ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG start_ARG ( 1 - italic_q ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_ARG ,

applying the inequality (1βˆ’x)⁒(1βˆ’y)β‰₯1βˆ’xβˆ’y1π‘₯1𝑦1π‘₯𝑦(1-x)(1-y)\geq 1-x-y( 1 - italic_x ) ( 1 - italic_y ) β‰₯ 1 - italic_x - italic_y for x,yβ©Ύ0π‘₯𝑦0x,y\geqslant 0italic_x , italic_y β©Ύ 0, we get

(3.2.4) 1βˆ’βˆi=0kβˆ’1(1βˆ’qnβˆ’i)(1βˆ’q)kβ©½βˆ‘i=0kβˆ’1qnβˆ’i(1βˆ’q)k=βˆ‘i=0kβˆ’1qβˆ’i(1βˆ’q)k⁒qn.1superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–superscript1π‘žπ‘˜superscriptsubscript𝑖0π‘˜1superscriptπ‘žπ‘›π‘–superscript1π‘žπ‘˜superscriptsubscript𝑖0π‘˜1superscriptπ‘žπ‘–superscript1π‘žπ‘˜superscriptπ‘žπ‘›\dfrac{1-\prod_{i=0}^{k-1}(1-q^{n-i})}{(1-q)^{k}}\leqslant\dfrac{\sum_{i=0}^{k% -1}q^{n-i}}{(1-q)^{k}}=\dfrac{\sum_{i=0}^{k-1}q^{-i}}{(1-q)^{k}}q^{n}.divide start_ARG 1 - ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG start_ARG ( 1 - italic_q ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_ARG β©½ divide start_ARG βˆ‘ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 - italic_q ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_ARG = divide start_ARG βˆ‘ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT italic_q start_POSTSUPERSCRIPT - italic_i end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 - italic_q ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_ARG italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT .

Hence, the upper bound for (3.2.2) is proportional to qnsuperscriptπ‘žπ‘›q^{n}italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, with the constant depending only on kπ‘˜kitalic_k.

Case 2: k1<ksubscriptπ‘˜1π‘˜k_{1}<kitalic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_k. In this case, expression (3.2.1) can be rewritten as

(3.2.5) q(n1βˆ’k1)⁒(kβˆ’k1)⁒(qn1;qβˆ’1)k1⁒|∏i=0kβˆ’k1βˆ’1(1βˆ’qnβˆ’n1βˆ’i)βˆ’βˆi=0kβˆ’1(1βˆ’qnβˆ’i)∏i=0kβˆ’1(1βˆ’qnβˆ’i)|.superscriptπ‘žsubscript𝑛1subscriptπ‘˜1π‘˜subscriptπ‘˜1subscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1subscriptπ‘˜1superscriptsubscriptproduct𝑖0π‘˜subscriptπ‘˜111superscriptπ‘žπ‘›subscript𝑛1𝑖superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–\displaystyle q^{(n_{1}-k_{1})(k-k_{1})}\left(q^{n_{1}};q^{-1}\right)_{k_{1}}% \left|\dfrac{\prod_{i=0}^{k-k_{1}-1}\left(1-q^{n-n_{1}-i}\right)-\prod_{i=0}^{% k-1}\left(1-q^{n-i}\right)}{\prod_{i=0}^{k-1}\left(1-q^{n-i}\right)}\right|.italic_q start_POSTSUPERSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | divide start_ARG ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_i end_POSTSUPERSCRIPT ) - ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG start_ARG ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG | .

The expression (3.2.5) depends on the sign of the difference in the numerator

(3.2.6) |∏i=0kβˆ’k1βˆ’1(1βˆ’qnβˆ’n1βˆ’i)βˆ’βˆi=0kβˆ’1(1βˆ’qnβˆ’i)|.superscriptsubscriptproduct𝑖0π‘˜subscriptπ‘˜111superscriptπ‘žπ‘›subscript𝑛1𝑖superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–\left|\prod_{i=0}^{k-k_{1}-1}\left(1-q^{n-n_{1}-i}\right)-\prod_{i=0}^{k-1}% \left(1-q^{n-i}\right)\right|.| ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_i end_POSTSUPERSCRIPT ) - ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) | .

If ∏i=0kβˆ’k1βˆ’1(1βˆ’qnβˆ’n1βˆ’i)βˆ’βˆi=0kβˆ’1(1βˆ’qnβˆ’i)β©Ύ0superscriptsubscriptproduct𝑖0π‘˜subscriptπ‘˜111superscriptπ‘žπ‘›subscript𝑛1𝑖superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–0\prod_{i=0}^{k-k_{1}-1}\left(1-q^{n-n_{1}-i}\right)-\prod_{i=0}^{k-1}\left(1-q% ^{n-i}\right)\geqslant 0∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_i end_POSTSUPERSCRIPT ) - ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) β©Ύ 0, then we have

(3.2.7) q(n1βˆ’k1)⁒(kβˆ’k1)⁒(qn1;qβˆ’1)k1⁒∏i=0kβˆ’k1βˆ’1(1βˆ’qnβˆ’n1βˆ’i)βˆ’βˆi=0kβˆ’1(1βˆ’qnβˆ’i)∏i=0kβˆ’1(1βˆ’qnβˆ’i),superscriptπ‘žsubscript𝑛1subscriptπ‘˜1π‘˜subscriptπ‘˜1subscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1subscriptπ‘˜1superscriptsubscriptproduct𝑖0π‘˜subscriptπ‘˜111superscriptπ‘žπ‘›subscript𝑛1𝑖superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–q^{(n_{1}-k_{1})(k-k_{1})}\left(q^{n_{1}};q^{-1}\right)_{k_{1}}\dfrac{\prod_{i% =0}^{k-k_{1}-1}\left(1-q^{n-n_{1}-i}\right)-\prod_{i=0}^{k-1}\left(1-q^{n-i}% \right)}{\prod_{i=0}^{k-1}\left(1-q^{n-i}\right)},italic_q start_POSTSUPERSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT divide start_ARG ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_i end_POSTSUPERSCRIPT ) - ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG start_ARG ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG ,

since q(n1βˆ’k1)⁒(kβˆ’k1)β©½1superscriptπ‘žsubscript𝑛1subscriptπ‘˜1π‘˜subscriptπ‘˜11q^{(n_{1}-k_{1})(k-k_{1})}\leqslant 1italic_q start_POSTSUPERSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT β©½ 1, ∏i=0kβˆ’k1βˆ’1(1βˆ’qnβˆ’n1βˆ’i)β©½1superscriptsubscriptproduct𝑖0π‘˜subscriptπ‘˜111superscriptπ‘žπ‘›subscript𝑛1𝑖1\prod_{i=0}^{k-k_{1}-1}\left(1-q^{n-n_{1}-i}\right)\leqslant 1∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_i end_POSTSUPERSCRIPT ) β©½ 1 the upper bound for (3.2.7) is

(qn1;qβˆ’1)k1⁒1βˆ’βˆi=0kβˆ’1(1βˆ’qnβˆ’i)∏i=0kβˆ’1(1βˆ’qnβˆ’i),subscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1subscriptπ‘˜11superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–\left(q^{n_{1}};q^{-1}\right)_{k_{1}}\dfrac{1-\prod_{i=0}^{k-1}\left(1-q^{n-i}% \right)}{\prod_{i=0}^{k-1}\left(1-q^{n-i}\right)},( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT divide start_ARG 1 - ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG start_ARG ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG ,

applying the same inequalities as in Case 1, we estimate the entire expression by  (3.2.4)

βˆ‘i=0kβˆ’1qβˆ’i(1βˆ’q)k⁒qn.superscriptsubscript𝑖0π‘˜1superscriptπ‘žπ‘–superscript1π‘žπ‘˜superscriptπ‘žπ‘›\dfrac{\sum_{i=0}^{k-1}q^{-i}}{(1-q)^{k}}q^{n}.divide start_ARG βˆ‘ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT italic_q start_POSTSUPERSCRIPT - italic_i end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 - italic_q ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_ARG italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT .

In the case when ∏i=0kβˆ’k1βˆ’1(1βˆ’qnβˆ’n1βˆ’i)βˆ’βˆi=0kβˆ’1(1βˆ’qnβˆ’i)<0superscriptsubscriptproduct𝑖0π‘˜subscriptπ‘˜111superscriptπ‘žπ‘›subscript𝑛1𝑖superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–0\prod_{i=0}^{k-k_{1}-1}\left(1-q^{n-n_{1}-i}\right)-\prod_{i=0}^{k-1}\left(1-q% ^{n-i}\right)<0∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_i end_POSTSUPERSCRIPT ) - ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) < 0, we write

(3.2.8) q(n1βˆ’k1)⁒(kβˆ’k1)⁒(qn1;qβˆ’1)k1⁒∏i=0kβˆ’1(1βˆ’qnβˆ’i)βˆ’βˆi=0kβˆ’k1βˆ’1(1βˆ’qnβˆ’n1βˆ’i)∏i=0kβˆ’1(1βˆ’qnβˆ’i),superscriptπ‘žsubscript𝑛1subscriptπ‘˜1π‘˜subscriptπ‘˜1subscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1subscriptπ‘˜1superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–superscriptsubscriptproduct𝑖0π‘˜subscriptπ‘˜111superscriptπ‘žπ‘›subscript𝑛1𝑖superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–q^{(n_{1}-k_{1})(k-k_{1})}\left(q^{n_{1}};q^{-1}\right)_{k_{1}}\dfrac{\prod_{i% =0}^{k-1}\left(1-q^{n-i}\right)-\prod_{i=0}^{k-k_{1}-1}\left(1-q^{n-n_{1}-i}% \right)}{\prod_{i=0}^{k-1}\left(1-q^{n-i}\right)},italic_q start_POSTSUPERSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT divide start_ARG ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) - ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_i end_POSTSUPERSCRIPT ) end_ARG start_ARG ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG ,

since ∏i=0kβˆ’1(1βˆ’qnβˆ’i)β©½1superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–1\prod_{i=0}^{k-1}\left(1-q^{n-i}\right)\leqslant 1∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) β©½ 1, the upper bound becomes

(3.2.9) q(n1βˆ’k1)⁒(kβˆ’k1)⁒(qn1;qβˆ’1)k1⁒1βˆ’βˆi=0kβˆ’k1βˆ’1(1βˆ’qnβˆ’n1βˆ’i)∏i=0kβˆ’1(1βˆ’qnβˆ’i),superscriptπ‘žsubscript𝑛1subscriptπ‘˜1π‘˜subscriptπ‘˜1subscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1subscriptπ‘˜11superscriptsubscriptproduct𝑖0π‘˜subscriptπ‘˜111superscriptπ‘žπ‘›subscript𝑛1𝑖superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–q^{(n_{1}-k_{1})(k-k_{1})}\left(q^{n_{1}};q^{-1}\right)_{k_{1}}\dfrac{1-\prod_% {i=0}^{k-k_{1}-1}\left(1-q^{n-n_{1}-i}\right)}{\prod_{i=0}^{k-1}\left(1-q^{n-i% }\right)},italic_q start_POSTSUPERSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT divide start_ARG 1 - ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_i end_POSTSUPERSCRIPT ) end_ARG start_ARG ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG ,

applying the same inequalities as in Case 1, we estimate the entire expression by

(3.2.10) q(n1βˆ’k1)⁒(kβˆ’k1)β’βˆ‘i=0kβˆ’k1βˆ’1qnβˆ’n1βˆ’i(1βˆ’q)k=βˆ‘i=0kβˆ’k1βˆ’1qn+n1⁒(kβˆ’k1βˆ’1)+k1⁒(k1βˆ’k)βˆ’i(1βˆ’q)k,superscriptπ‘žsubscript𝑛1subscriptπ‘˜1π‘˜subscriptπ‘˜1superscriptsubscript𝑖0π‘˜subscriptπ‘˜11superscriptπ‘žπ‘›subscript𝑛1𝑖superscript1π‘žπ‘˜superscriptsubscript𝑖0π‘˜subscriptπ‘˜11superscriptπ‘žπ‘›subscript𝑛1π‘˜subscriptπ‘˜11subscriptπ‘˜1subscriptπ‘˜1π‘˜π‘–superscript1π‘žπ‘˜q^{(n_{1}-k_{1})(k-k_{1})}\frac{\sum_{i=0}^{k-k_{1}-1}q^{n-n_{1}-i}}{(1-q)^{k}% }=\dfrac{\sum_{i=0}^{k-k_{1}-1}q^{n+n_{1}(k-k_{1}-1)+k_{1}(k_{1}-k)-i}}{(1-q)^% {k}},italic_q start_POSTSUPERSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT divide start_ARG βˆ‘ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT italic_q start_POSTSUPERSCRIPT italic_n - italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_i end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 - italic_q ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_ARG = divide start_ARG βˆ‘ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT italic_q start_POSTSUPERSCRIPT italic_n + italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 ) + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k ) - italic_i end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 - italic_q ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_ARG ,

since kβˆ’k1βˆ’1β©Ύ0π‘˜subscriptπ‘˜110k-k_{1}-1\geqslant 0italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 β©Ύ 0, we obtain a uniform bound with respect to the parameter n1subscript𝑛1n_{1}italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT

(3.2.11) βˆ‘i=0kβˆ’k1βˆ’1qn+n1⁒(kβˆ’k1βˆ’1)+k1⁒(k1βˆ’k)βˆ’i(1βˆ’q)kβ©½βˆ‘i=0kβˆ’k1βˆ’1qk1⁒(k1βˆ’k)βˆ’i(1βˆ’q)k⁒qn.superscriptsubscript𝑖0π‘˜subscriptπ‘˜11superscriptπ‘žπ‘›subscript𝑛1π‘˜subscriptπ‘˜11subscriptπ‘˜1subscriptπ‘˜1π‘˜π‘–superscript1π‘žπ‘˜superscriptsubscript𝑖0π‘˜subscriptπ‘˜11superscriptπ‘žsubscriptπ‘˜1subscriptπ‘˜1π‘˜π‘–superscript1π‘žπ‘˜superscriptπ‘žπ‘›\dfrac{\sum_{i=0}^{k-k_{1}-1}q^{n+n_{1}(k-k_{1}-1)+k_{1}(k_{1}-k)-i}}{(1-q)^{k% }}\leqslant\dfrac{\sum_{i=0}^{k-k_{1}-1}q^{k_{1}(k_{1}-k)-i}}{(1-q)^{k}}q^{n}.divide start_ARG βˆ‘ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT italic_q start_POSTSUPERSCRIPT italic_n + italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 ) + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k ) - italic_i end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 - italic_q ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_ARG β©½ divide start_ARG βˆ‘ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT italic_q start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k ) - italic_i end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 - italic_q ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_ARG italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT .

In each of the two cases, the upper bound is of order qnsuperscriptπ‘žπ‘›q^{n}italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, with the constant depending only on kπ‘˜kitalic_k and k1subscriptπ‘˜1k_{1}italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT.

Combining the two cases, we conclude that the overall bound is of the form ckβ‹…qnβ‹…subscriptπ‘π‘˜superscriptπ‘žπ‘›c_{k}\cdot q^{n}italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT β‹… italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, where cksubscriptπ‘π‘˜c_{k}italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is a constant depending only on kπ‘˜kitalic_k. This completes the proof of the lemma.

∎

3.3. From finite to infinite

Since the set Ξ”qsubscriptΞ”π‘ž\Delta_{q}roman_Ξ” start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT is compact, the probability measures on Ξ”qsubscriptΞ”π‘ž\Delta_{q}roman_Ξ” start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT are uniquely determined by sequences of their moments. Therefore, injectivity of the map  (2.1.7) is automatic. Using Theorem  (3.1.1), we prove the surjectivity of the map  (2.1.7), thereby rederiving the result of Gnedin–Olshanski  (2.1.7).

Corollary 3.3.1.

The map  (2.1.7) is surjective.

Proof.

Let β„™β„™\mathbb{P}roman_β„™ be a qπ‘žqitalic_q-exchangeable probability measure on {0,1}∞superscript01\{0,1\}^{\infty}{ 0 , 1 } start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT. Consider the natural projections β„™nsubscriptℙ𝑛\mathbb{P}_{n}roman_β„™ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT onto {0,1}nsuperscript01𝑛\{0,1\}^{n}{ 0 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. From Theorem (3.1.1) we obtain a family of measures ΞΌnsubscriptπœ‡π‘›\mu_{n}italic_ΞΌ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. By compactness of the space of probability measures on [0,1]01[0,1][ 0 , 1 ], we can extract subsequence ΞΌnisubscriptπœ‡subscript𝑛𝑖\mu_{n_{i}}italic_ΞΌ start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT that converges weakly to a probability measure ΞΌπœ‡\muitalic_ΞΌ. Consequently, we obtain the weak convergence β„™ΞΌni,kβ†’niβ†’βˆžweaklyβ„™ΞΌ,k.β†’subscript𝑛𝑖weaklyβ†’subscriptβ„™subscriptπœ‡subscriptπ‘›π‘–π‘˜subscriptβ„™πœ‡π‘˜\mathbb{P}_{\mu_{n_{i}},k}\xrightarrow[n_{i}\to\infty]{\text{weakly}}\mathbb{P% }_{\mu,k}.roman_β„™ start_POSTSUBSCRIPT italic_ΞΌ start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_k end_POSTSUBSCRIPT start_ARROW start_UNDERACCENT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT β†’ ∞ end_UNDERACCENT start_ARROW overweakly β†’ end_ARROW end_ARROW roman_β„™ start_POSTSUBSCRIPT italic_ΞΌ , italic_k end_POSTSUBSCRIPT . Since β€–β„™ΞΌni,kβˆ’β„™kβ€–β†’niβ†’βˆž0β†’subscript𝑛𝑖absentβ†’normsubscriptβ„™subscriptπœ‡subscriptπ‘›π‘–π‘˜subscriptβ„™π‘˜0\|\mathbb{P}_{\mu_{n_{i}},k}-\mathbb{P}_{k}\|\xrightarrow[n_{i}\to\infty]{}0βˆ₯ roman_β„™ start_POSTSUBSCRIPT italic_ΞΌ start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_k end_POSTSUBSCRIPT - roman_β„™ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT βˆ₯ start_ARROW start_UNDERACCENT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT β†’ ∞ end_UNDERACCENT start_ARROW start_OVERACCENT end_OVERACCENT β†’ end_ARROW end_ARROW 0, we have β„™ΞΌ,k=β„™ksubscriptβ„™πœ‡π‘˜subscriptβ„™π‘˜{\mathbb{P}}_{\mu,k}={\mathbb{P}}_{k}roman_β„™ start_POSTSUBSCRIPT italic_ΞΌ , italic_k end_POSTSUBSCRIPT = roman_β„™ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT for all kπ‘˜kitalic_k. We conclude that β„™=β„™ΞΌ=βˆ«Ξ”qΞ½xq⁒μ⁒(d⁒x)β„™subscriptβ„™πœ‡subscriptsubscriptΞ”π‘žsubscriptsuperscriptπœˆπ‘žπ‘₯πœ‡π‘‘π‘₯\mathbb{P}=\mathbb{P}_{\mu}=\displaystyle\int_{\Delta_{q}}\nu^{q}_{x}\,\mu(dx)roman_β„™ = roman_β„™ start_POSTSUBSCRIPT italic_ΞΌ end_POSTSUBSCRIPT = ∫ start_POSTSUBSCRIPT roman_Ξ” start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_ΞΌ ( italic_d italic_x ).

∎

3.4. The rate is sharp

We provide an example in which the lower bound for the variational distance in Theorem (3.1.1) is of order qnsuperscriptπ‘žπ‘›q^{n}italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, confirming that this rate is optimal. The example is given by the extreme measure en,n1qsubscriptsuperscriptπ‘’π‘žπ‘›subscript𝑛1e^{q}_{n,n_{1}}italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT and the measure Ξ½xqsubscriptsuperscriptπœˆπ‘žπ‘₯\nu^{q}_{x}italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT with parameter x=qn1π‘₯superscriptπ‘žsubscript𝑛1x=q^{n_{1}}italic_x = italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT. We begin by proving a technical lemma.

Lemma 3.4.1.
(3.4.2) 1βˆ’βˆi=0kβˆ’1(1βˆ’qnβˆ’i)∏i=0kβˆ’1(1βˆ’qnβˆ’i)β©Ύq1βˆ’kβˆ’q1βˆ’q⁒qn.1superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–superscriptπ‘ž1π‘˜π‘ž1π‘žsuperscriptπ‘žπ‘›\dfrac{1-\prod_{i=0}^{k-1}(1-q^{n-i})}{\prod_{i=0}^{k-1}(1-q^{n-i})}\geqslant% \dfrac{q^{1-k}-q}{1-q}q^{n}.divide start_ARG 1 - ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG start_ARG ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG β©Ύ divide start_ARG italic_q start_POSTSUPERSCRIPT 1 - italic_k end_POSTSUPERSCRIPT - italic_q end_ARG start_ARG 1 - italic_q end_ARG italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT .
Proof.

Since ln⁑(1βˆ’x)β©½βˆ’x1π‘₯π‘₯\ln(1-x)\leqslant-xroman_ln ( 1 - italic_x ) β©½ - italic_x for x∈(0,1)π‘₯01x\in(0,1)italic_x ∈ ( 0 , 1 ), we have

(3.4.3) ∏i=0kβˆ’1(1βˆ’qnβˆ’i)β©½exp⁑(βˆ’βˆ‘i=0kβˆ’1qnβˆ’i),superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–superscriptsubscript𝑖0π‘˜1superscriptπ‘žπ‘›π‘–\prod_{i=0}^{k-1}(1-q^{n-i})\leqslant\exp\left(-\sum_{i=0}^{k-1}q^{n-i}\right),∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) β©½ roman_exp ( - βˆ‘ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) ,

therefore,

(3.4.4) 1βˆ’βˆi=0kβˆ’1(1βˆ’qnβˆ’i)∏i=0kβˆ’1(1βˆ’qnβˆ’i)=1∏i=0kβˆ’1(1βˆ’qnβˆ’i)βˆ’1β©Ύexp⁑(βˆ‘i=0kβˆ’1qnβˆ’i)βˆ’1.1superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–1superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–1superscriptsubscript𝑖0π‘˜1superscriptπ‘žπ‘›π‘–1\dfrac{1-\prod_{i=0}^{k-1}(1-q^{n-i})}{\prod_{i=0}^{k-1}(1-q^{n-i})}=\dfrac{1}% {\prod_{i=0}^{k-1}(1-q^{n-i})}-1\geqslant\exp\left(\sum_{i=0}^{k-1}q^{n-i}% \right)-1.divide start_ARG 1 - ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG start_ARG ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG = divide start_ARG 1 end_ARG start_ARG ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG - 1 β©Ύ roman_exp ( βˆ‘ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) - 1 .

Since exp⁑(x)βˆ’1β©Ύxπ‘₯1π‘₯\exp(x)-1\geqslant xroman_exp ( italic_x ) - 1 β©Ύ italic_x for xβ©Ύ0π‘₯0x\geqslant 0italic_x β©Ύ 0, it follows that

(3.4.5) 1βˆ’βˆi=0kβˆ’1(1βˆ’qnβˆ’i)∏i=0kβˆ’1(1βˆ’qnβˆ’i)β©Ύβˆ‘i=0kβˆ’1qnβˆ’i=q1βˆ’kβˆ’q1βˆ’q⁒qn.1superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–superscriptsubscript𝑖0π‘˜1superscriptπ‘žπ‘›π‘–superscriptπ‘ž1π‘˜π‘ž1π‘žsuperscriptπ‘žπ‘›\dfrac{1-\prod_{i=0}^{k-1}(1-q^{n-i})}{\prod_{i=0}^{k-1}(1-q^{n-i})}\geqslant% \sum_{i=0}^{k-1}q^{n-i}=\dfrac{q^{1-k}-q}{1-q}q^{n}.divide start_ARG 1 - ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG start_ARG ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG β©Ύ βˆ‘ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT = divide start_ARG italic_q start_POSTSUPERSCRIPT 1 - italic_k end_POSTSUPERSCRIPT - italic_q end_ARG start_ARG 1 - italic_q end_ARG italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT .

∎

Proposition 3.4.6.

For n1β©Ύksubscript𝑛1π‘˜n_{1}\geqslant kitalic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT β©Ύ italic_k, we have

(3.4.7) β€–(en,n1q)kβˆ’(Ξ½qn1q)kβ€–β©Ύc~kβ‹…qn,normsubscriptsubscriptsuperscriptπ‘’π‘žπ‘›subscript𝑛1π‘˜subscriptsubscriptsuperscriptπœˆπ‘žsuperscriptπ‘žsubscript𝑛1π‘˜β‹…subscript~π‘π‘˜superscriptπ‘žπ‘›\big{\|}(e^{q}_{n,n_{1}})_{k}-(\nu^{q}_{q^{n_{1}}})_{k}\big{\|}\geqslant\tilde% {c}_{k}\cdot q^{n},βˆ₯ ( italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - ( italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT βˆ₯ β©Ύ over~ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT β‹… italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ,

where c~ksubscript~π‘π‘˜\tilde{c}_{k}over~ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is a constant depending only on kπ‘˜kitalic_k.

Proof.

As we have already shown, the variational distance between en,n1qsubscriptsuperscriptπ‘’π‘žπ‘›subscript𝑛1e^{q}_{n,n_{1}}italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT and Ξ½xqsubscriptsuperscriptπœˆπ‘žπ‘₯\nu^{q}_{x}italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT can be computed using the following formula

β€–(en,n1q)kβˆ’(Ξ½qn1q)kβ€–=βˆ‘k1=0k[kk1]⁒q(n1βˆ’k1)⁒(kβˆ’k1)⁒|[nβˆ’kn1βˆ’k1]/[nn1]βˆ’(qn1;qβˆ’1)k1|.normsubscriptsubscriptsuperscriptπ‘’π‘žπ‘›subscript𝑛1π‘˜subscriptsubscriptsuperscriptπœˆπ‘žsuperscriptπ‘žsubscript𝑛1π‘˜superscriptsubscriptsubscriptπ‘˜10π‘˜FRACOPπ‘˜subscriptπ‘˜1superscriptπ‘žsubscript𝑛1subscriptπ‘˜1π‘˜subscriptπ‘˜1FRACOPπ‘›π‘˜subscript𝑛1subscriptπ‘˜1FRACOP𝑛subscript𝑛1subscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1subscriptπ‘˜1\big{\|}(e^{q}_{n,n_{1}})_{k}-(\nu^{q}_{q^{n_{1}}})_{k}\big{\|}=\sum_{k_{1}=0}% ^{k}\genfrac{[}{]}{0.0pt}{}{k}{k_{1}}\,q^{(n_{1}-k_{1})(k-k_{1})}\left|% \genfrac{[}{]}{0.0pt}{}{n-k}{n_{1}-k_{1}}\bigg{/}\genfrac{[}{]}{0.0pt}{}{n}{n_% {1}}-(q^{n_{1}};q^{-1})_{k_{1}}\right|.βˆ₯ ( italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - ( italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT βˆ₯ = βˆ‘ start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT [ FRACOP start_ARG italic_k end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ] italic_q start_POSTSUPERSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_k - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT | [ FRACOP start_ARG italic_n - italic_k end_ARG start_ARG italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ] / [ FRACOP start_ARG italic_n end_ARG start_ARG italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ] - ( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | .

Since n1β©Ύksubscript𝑛1π‘˜n_{1}\geqslant kitalic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT β©Ύ italic_k, we have (qn1;qβˆ’1)kβ‰ 0subscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1π‘˜0(q^{n_{1}};q^{-1})_{k}\neq 0( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT β‰  0. To obtain a lower bound, we consider only the term corresponding to k1=ksubscriptπ‘˜1π‘˜k_{1}=kitalic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_k in the sum.

β€–(en,n1q)kβˆ’(Ξ½qn1q)kβ€–β©Ύ(qn1;qβˆ’1)k⁒1βˆ’βˆi=0kβˆ’1(1βˆ’qnβˆ’i)∏i=0kβˆ’1(1βˆ’qnβˆ’i).normsubscriptsubscriptsuperscriptπ‘’π‘žπ‘›subscript𝑛1π‘˜subscriptsubscriptsuperscriptπœˆπ‘žsuperscriptπ‘žsubscript𝑛1π‘˜subscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1π‘˜1superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–\big{\|}(e^{q}_{n,n_{1}})_{k}-(\nu^{q}_{q^{n_{1}}})_{k}\big{\|}\geqslant(q^{n_% {1}};q^{-1})_{k}\dfrac{1-\prod_{i=0}^{k-1}(1-q^{n-i})}{\prod_{i=0}^{k-1}(1-q^{% n-i})}.βˆ₯ ( italic_e start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - ( italic_Ξ½ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT βˆ₯ β©Ύ ( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT divide start_ARG 1 - ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG start_ARG ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG .

Using the inequality (qn1;qβˆ’1)kβ©Ύ(1βˆ’q)ksubscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1π‘˜superscript1π‘žπ‘˜(q^{n_{1}};q^{-1})_{k}\geqslant(1-q)^{k}( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT β©Ύ ( 1 - italic_q ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT and Lemma (3.4.1), we obtain

(3.4.8) (qn1;qβˆ’1)k⁒1βˆ’βˆi=0kβˆ’1(1βˆ’qnβˆ’i)∏i=0kβˆ’1(1βˆ’qnβˆ’i)β©Ύ(1βˆ’q)k⁒q1βˆ’kβˆ’q1βˆ’q⁒qn.subscriptsuperscriptπ‘žsubscript𝑛1superscriptπ‘ž1π‘˜1superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–superscriptsubscriptproduct𝑖0π‘˜11superscriptπ‘žπ‘›π‘–superscript1π‘žπ‘˜superscriptπ‘ž1π‘˜π‘ž1π‘žsuperscriptπ‘žπ‘›(q^{n_{1}};q^{-1})_{k}\dfrac{1-\prod_{i=0}^{k-1}(1-q^{n-i})}{\prod_{i=0}^{k-1}% (1-q^{n-i})}\geqslant(1-q)^{k}\dfrac{q^{1-k}-q}{1-q}q^{n}.( italic_q start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ; italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT divide start_ARG 1 - ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG start_ARG ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( 1 - italic_q start_POSTSUPERSCRIPT italic_n - italic_i end_POSTSUPERSCRIPT ) end_ARG β©Ύ ( 1 - italic_q ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT divide start_ARG italic_q start_POSTSUPERSCRIPT 1 - italic_k end_POSTSUPERSCRIPT - italic_q end_ARG start_ARG 1 - italic_q end_ARG italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT .

Thus, we see that the lower bound is of order qnsuperscriptπ‘žπ‘›q^{n}italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. ∎

References

  • [BO16] Alexei Borodin and Grigori Olshanski. Representations of the Infinite Symmetric Group. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2016.
  • [DF80] P. Diaconis and D. Freedman. Finite Exchangeable Sequences. The Annals of Probability, 8(4):745 – 764, 1980.
  • [Fel71] W. Feller. An Introduction to Probability Theory and Its Application Vol II. John Wiley and Sons, 1971.
  • [GO09] Alexander Gnedin and Grigori Olshanski. A q-analogue of de Finetti’s theorem. The Electronic Journal of Combinatorics, Volume 16, Issue 1 (2009).
  • [GO10] Alexander Gnedin and Grigori Olshanski. q-exchangeability via quasi-invariance. The Annals of Probability, 38(6), November 2010.
  • [HS55] Edwin Hewitt and Leonard J. Savage. Symmetric Measures on Cartesian Products. Transactions of the American Mathematical Society, 80(2):470–501, 1955.
  • [Kir18] Werner Kirsch. An elementary proof of de Finetti’s Theorem, 2018.