Title: A doubly composite Chernoff-Stein lemma and its applications Show Chinese title

Title: 一个双重复合Chernoff-Stein引理及其应用

Abstract: Given a sequence of random variables $X^n=X_1,\ldots, X_n$, discriminating between two hypotheses on the underlying probability distribution is a key task in statistics and information theory. Of interest here is the Stein exponent, i.e. the largest rate of decay (in $n$) of the type II error probability for a vanishingly small type I error probability. When the hypotheses are simple and i.i.d., the Chernoff-Stein lemma states that this is given by the relative entropy between the single-copy probability distributions. Generalisations of this result exist in the case of composite hypotheses, but mostly to settings where the probability distribution of $X^n$ is not genuinely correlated, but rather, e.g., a convex combination of product distributions with components taken from a base set. Here, we establish a general Chernoff-Stein lemma that applies to the setting where both hypotheses are composite and genuinely correlated, satisfying only generic assumptions such as convexity (on both hypotheses) and some weak form of permutational symmetry (on either hypothesis). Our result, which strictly subsumes most prior work, is proved using a refinement of the blurring technique developed in the context of the generalised quantum Stein's lemma [Lami, IEEE Trans. Inf. Theory 2025]. In this refined form, blurring is applied symbol by symbol, which makes it both stronger and applicable also in the absence of permutational symmetry. The second part of the work is devoted to applications: we provide a single-letter formula for the Stein exponent characterising the discrimination of broad families of null hypotheses vs a composite i.i.d. or an arbitrarily varying alternative hypothesis, and establish a 'constrained de Finetti reduction' statement that covers a wide family of convex constraints. Applications to quantum hypothesis testing are explored in a related paper [Lami, arXiv:today]. Show Chinese abstract

Abstract: 给定一个随机变量序列$X^n=X_1,\ldots, X_n$，在基础概率分布上区分两个假设是统计学和信息论中的关键任务。 此处关注的是Stein指数，即当类型I错误概率趋于零时，类型II错误概率在$n$方面的最大衰减率。 当假设是简单且独立同分布时，Chernoff-Stein引理指出，这由单个副本概率分布之间的相对熵给出。 在复合假设的情况下，这一结果存在推广，但主要适用于$X^n$的概率分布并非真正相关，而是例如从基本集合中选取的乘积分布的凸组合的情况。 在此工作中，我们建立了一个一般的Chernoff-Stein引理，适用于两个假设都是复合且真正相关的场景，仅需满足一些通用假设，如凸性（在两个假设上）以及某种弱形式的排列对称性（在一个假设上）。 我们的结果严格涵盖了大多数先前的工作，其证明使用了在广义量子Stein引理背景下开发的模糊技术的改进版本[Lami, IEEE Trans. Inf. Theory 2025]。 在这种改进形式下，模糊技术逐个符号应用，使其更强，并且在没有排列对称性的条件下也适用。 工作的第二部分致力于应用：我们提供了一个单字母公式，用于表征区分广泛的零假设与复合独立同分布或任意变化的替代假设的Stein指数，并建立了一个“受限的de Finetti约简”陈述，涵盖了一大类凸约束。 在相关论文[Lami, arXiv:today]中探讨了量子假设检验的应用。