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
SN - 0304-4149
VL - 130
SP - 2250
EP - 2281
JO - Stochastic Processes and their Applications
JF - Stochastic Processes and their Applications
IS - 4
ER -