A CLT for linear spectral statistics of large random information-plus-noise matrices

Marwa Banna; Jamal Najim; Jianfeng Yao

doi:10.1016/j.spa.2019.06.017

A CLT for linear spectral statistics of large random information-plus-noise matrices

Marwa Banna, Jamal Najim, Jianfeng Yao

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Consider a matrix [Formula presented] where σ>0 and X_n=(x_ij ⁿ) is a N×n random matrix with i.i.d. real or complex standardized entries and A_n is a N×n deterministic matrix with bounded spectral norm. The fluctuations of the linear spectral statistics of the eigenvalues: Tracef(Y_nY_n ^∗)=∑i=1Nf(λ_i),(λ_i)eigenvalues ofY_nY_n ^∗,are shown to be Gaussian, in the case where f is a smooth function of class C³ with bounded support, and in the regime where both dimensions of matrix Y_n go to infinity at the same pace. The CLT is expressed in terms of vanishing Lévy–Prokhorov distance between the linear statistics’ distribution and a centered Gaussian probability distribution, the variance of which depends upon N and n and may not converge. The proof combines ideas from [2,18] and [32].

Original language	English (US)
Pages (from-to)	2250-2281
Number of pages	32
Journal	Stochastic Processes and their Applications
Volume	130
Issue number	4
DOIs	https://doi.org/10.1016/j.spa.2019.06.017
State	Published - Apr 2020

Keywords

Central limit theorem
Large random matrices
Linear statistics of the eigenvalues

ASJC Scopus subject areas

Statistics and Probability
Modeling and Simulation
Applied Mathematics

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10.1016/j.spa.2019.06.017

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title = "A CLT for linear spectral statistics of large random information-plus-noise matrices",

abstract = "Consider a matrix [Formula presented] where σ>0 and Xn=(xij n) is a N×n random matrix with i.i.d. real or complex standardized entries and An is a N×n deterministic matrix with bounded spectral norm. The fluctuations of the linear spectral statistics of the eigenvalues: Tracef(YnYn ∗)=∑i=1Nf(λi),(λi)eigenvalues ofYnYn ∗,are shown to be Gaussian, in the case where f is a smooth function of class C3 with bounded support, and in the regime where both dimensions of matrix Yn go to infinity at the same pace. The CLT is expressed in terms of vanishing L{\'e}vy–Prokhorov distance between the linear statistics{\textquoteright} distribution and a centered Gaussian probability distribution, the variance of which depends upon N and n and may not converge. The proof combines ideas from [2,18] and [32].",

keywords = "Central limit theorem, Large random matrices, Linear statistics of the eigenvalues",

author = "Marwa Banna and Jamal Najim and Jianfeng Yao",

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AU - Banna, Marwa

AU - Najim, Jamal

AU - Yao, Jianfeng

PY - 2020/4

Y1 - 2020/4

N2 - Consider a matrix [Formula presented] where σ>0 and Xn=(xij n) is a N×n random matrix with i.i.d. real or complex standardized entries and An is a N×n deterministic matrix with bounded spectral norm. The fluctuations of the linear spectral statistics of the eigenvalues: Tracef(YnYn ∗)=∑i=1Nf(λi),(λi)eigenvalues ofYnYn ∗,are shown to be Gaussian, in the case where f is a smooth function of class C3 with bounded support, and in the regime where both dimensions of matrix Yn go to infinity at the same pace. The CLT is expressed in terms of vanishing Lévy–Prokhorov distance between the linear statistics’ distribution and a centered Gaussian probability distribution, the variance of which depends upon N and n and may not converge. The proof combines ideas from [2,18] and [32].

AB - Consider a matrix [Formula presented] where σ>0 and Xn=(xij n) is a N×n random matrix with i.i.d. real or complex standardized entries and An is a N×n deterministic matrix with bounded spectral norm. The fluctuations of the linear spectral statistics of the eigenvalues: Tracef(YnYn ∗)=∑i=1Nf(λi),(λi)eigenvalues ofYnYn ∗,are shown to be Gaussian, in the case where f is a smooth function of class C3 with bounded support, and in the regime where both dimensions of matrix Yn go to infinity at the same pace. The CLT is expressed in terms of vanishing Lévy–Prokhorov distance between the linear statistics’ distribution and a centered Gaussian probability distribution, the variance of which depends upon N and n and may not converge. The proof combines ideas from [2,18] and [32].

KW - Central limit theorem

KW - Large random matrices

KW - Linear statistics of the eigenvalues

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A CLT for linear spectral statistics of large random information-plus-noise matrices

Abstract

Keywords

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