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

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  • Banna, Marwa
  • Najim, Jamal
  • Yao, Jianfeng

Abstract

Consider a matrix Yn=σnXn+An, where σ>0 and Xn=(xijn) 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].

Suggested Citation

  • Banna, Marwa & Najim, Jamal & Yao, Jianfeng, 2020. "A CLT for linear spectral statistics of large random information-plus-noise matrices," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2250-2281.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:4:p:2250-2281
    DOI: 10.1016/j.spa.2019.06.017
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    References listed on IDEAS

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    1. Dozier, R. Brent & Silverstein, Jack W., 2007. "Analysis of the limiting spectral distribution of large dimensional information-plus-noise type matrices," Journal of Multivariate Analysis, Elsevier, vol. 98(6), pages 1099-1122, July.
    2. Dozier, R. Brent & Silverstein, Jack W., 2007. "On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices," Journal of Multivariate Analysis, Elsevier, vol. 98(4), pages 678-694, April.
    3. Hachem, Walid & Loubaton, Philippe & Mestre, Xavier & Najim, Jamal & Vallet, Pascal, 2013. "A subspace estimator for fixed rank perturbations of large random matrices," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 427-447.
    4. Jonsson, Dag, 1982. "Some limit theorems for the eigenvalues of a sample covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 12(1), pages 1-38, March.
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