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Functional CLT for general sample covariance matrices

Jian Cui, Zhijun Liu, Jiang Hu, Zhidong Bai

Abstract

This paper studies the central limit theorems (CLTs) for linear spectral statistics (LSSs) of general sample covariance matrices, when the test functions belong to $C^3$, the class of functions with continuous third order derivatives. We consider matrices of the form $B_n=(1/n)T_p^{1/2}X_nX_n^{*}T_p^{1/2},$ where $X_n= (x_{i j} ) $ is a $p \times n$ matrix whose entries are independent and identically distributed (i.i.d.) real or complex random variables, and $T_p$ is a $p\times p$ nonrandom Hermitian nonnegative definite matrix with its spectral norm uniformly bounded in $p$. By using Bernstein polynomial approximation, we show that, under $\mathbb{E}|x_{ij}|^{8}<\infty$, the centered LSSs of $B_n$ have Gaussian limits. Under the stronger $\mathbb{E}|x_{ij}|^{10}<\infty$, we further establish convergence rates $O(n^{-1/2+κ})$ in Kolmogorov--Smirnov $O(n^{-1/2+κ})$, for any fixed $κ>0$.

Functional CLT for general sample covariance matrices

Abstract

This paper studies the central limit theorems (CLTs) for linear spectral statistics (LSSs) of general sample covariance matrices, when the test functions belong to , the class of functions with continuous third order derivatives. We consider matrices of the form where is a matrix whose entries are independent and identically distributed (i.i.d.) real or complex random variables, and is a nonrandom Hermitian nonnegative definite matrix with its spectral norm uniformly bounded in . By using Bernstein polynomial approximation, we show that, under , the centered LSSs of have Gaussian limits. Under the stronger , we further establish convergence rates in Kolmogorov--Smirnov , for any fixed .
Paper Structure (15 sections, 21 theorems, 218 equations, 1 figure)

This paper contains 15 sections, 21 theorems, 218 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Literature review
  4. Organization
  5. Notation and conventions
  6. Contribution and proof strategy
  7. Bernstein polynomial approximations
  8. Proof of Theorem \ref{['FunctionCLT']}
  9. Truncation and normalization
  10. Stieltjes transformation and LSS
  11. Random part
  12. Nonrandom part
  13. Proof of Theorem \ref{['RateFunctionCLT']}
  14. Useful lemmas
  15. Proof of (\ref{['Qjdecom']}).

Key Result

Theorem 1.1

$(i)$ Under Assumptions assum8th--assumRG and assumhadma, then the random vector converges weakly to a Gaussian vector $(X_{f_1},\dots,X_{f_k})$, with mean and covariance function where The contours $\mathcal{C},\mathcal{C}_1,\mathcal{C}_2$ are closed and oriented in the positive direction in the complex plane, each enclosing the support set defined in supportset. We may assume $\mathcal{C}_1$

Figures (1)

  • Figure 3.1: Contour $\mathcal{C}$.

Theorems & Definitions (39)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.1
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.2
  • Remark 1.5
  • Remark 1.6
  • proof : Proof of Lyapunov condition
  • proof : Proof of conditional covariance
  • ...and 29 more