TY - JOUR

T1 - A CLT for linear spectral statistics of large random information-plus-noise matrices

AU - Banna, Marwa

AU - Najim, Jamal

AU - Yao, Jianfeng

N1 - Funding Information:
MB is supported by the ERC Advanced Grant NCDFP339760, held by Roland Speicher, and partially by the French ANR grant ANR-12-MONU-0003.JN is supported by French ANR grant ANR-12-MONU-0003 and Labex Bézout.JY is supported by program “Futuret ruptures” of Fondation Télécom.
Funding Information:
MB is supported by the ERC Advanced Grant NCDFP339760, held by Roland Speicher, and partially by the French ANR grant ANR-12-MONU-0003.JN is supported by French ANR grant ANR-12-MONU-0003 and Labex B?zout.JY is supported by program ?Futuret ruptures? of Fondation T?l?com.
Publisher Copyright:
© 2019 Elsevier B.V.

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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U2 - 10.1016/j.spa.2019.06.017

DO - 10.1016/j.spa.2019.06.017

M3 - Article

AN - SCOPUS:85068525176

VL - 130

SP - 2250

EP - 2281

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 4

ER -