Title: Spectral estimation for high-dimensional linear processes Show Chinese title

Title: 高维线性过程的谱估计

Abstract: We propose a novel estimation procedure for certain spectral distributions associated with a class of high dimensional linear time series. The processes under consideration are of the form $X_t = \sum_{\ell=0}^\infty \mathbf{A}_\ell Z_{t-\ell}$ with iid innovations $(Z_t)$. The key structural assumption is that the coefficient matrices and the variance of the innovations are simultaneously diagonalizable in a common orthonormal basis. We develop a strategy for estimating the joint spectral distribution of the coefficient matrices and the innovation variance by making use of the asymptotic behavior of the eigenvalues of appropriately weighted integrals of the sample periodogram. Throughout we work under the asymptotic regime $p,n \to \infty$, such that $p/n\to c \in (0,\infty)$, where $p$ is the dimension and $n$ is the sample size. Under this setting, we first establish a weak limit for the empirical distribution of eigenvalues of the aforementioned integrated sample periodograms. This result is proved using techniques from random matrix theory, in particular the characterization of weak convergence by means of the Stieltjes transform of relevant distributions. We utilize this result to develop an estimator of the joint spectral distribution of the coefficient matrices, by minimizing an $L^\kappa$ discrepancy measure, for $\kappa \geq 1$, between the empirical and limiting Stieltjes transforms of the integrated sample periodograms. This is accomplished by assuming that the joint spectral distribution is a discrete mixture of point masses. We also prove consistency of the estimator corresponding to the $L^2$ discrepancy measure. We illustrate the methodology through simulations and an application to stock price data from the S\&P 500 series. Show Chinese abstract

Abstract: 我们提出了一种新颖的估计方法，用于一类高维线性时间序列相关的某些谱分布。 所考虑的过程形式为 $X_t = \sum_{\ell=0}^\infty \mathbf{A}_\ell Z_{t-\ell}$，其中创新项为独立同分布（iid）的 $(Z_t)$。 关键的结构假设是系数矩阵和创新项的方差可以在一个共同的规范正交基下同时对角化。 我们通过利用适当加权积分的样本周期图的特征值的渐近行为，发展了一种策略来估计系数矩阵和创新方差的联合谱分布。 在整个过程中，我们采用渐近框架 $p,n \to \infty$，使得 $p/n\to c \in (0,\infty)$，其中 $p$为维度， $n$为样本大小。 在此设定下，我们首先建立了上述集成样本周期图特征值经验分布的弱极限。 这一结果通过随机矩阵理论的技术证明，特别是利用相关分布的Stieltjes变换表征弱收敛的方法。 我们利用这一结果开发了一种系数矩阵联合谱分布的估计量，通过最小化 $L^\kappa$ 在 $\kappa \geq 1$ 处的经验与极限Stieltjes变换之间的差异度量。这是通过假设联合谱分布是一些点质量的离散混合来实现的。我们还证明了与 $L^2$ 差异度量对应的估计量的一致性。我们通过模拟以及对标准普尔500指数股票价格数据的应用来展示该方法论。