IUMJ

Title: Exact separation phenomenon for the eigenvalues of large Information-Plus-Noise type matrices. Application to spiked models

Authors: Mireille Capitaine

Issue: Volume 63 (2014), Issue 6, 1875-1910

Abstract:

We consider large Information-plus-noise type matrices of the form $M_N=(\sigma X_N/\sqrt{N}+A_N)(\sigma X_N/\sqrt{N}+A_N)^{*}$ where $X_N$ is an $n\times N$ ($n\leq N)$ matrix consisting of independent standardized complex entries, $A_N$ is an $n\times N$ nonrandom matrix, and $\sigma>0$. As $N$ tends to infinity, if $n/N\to c\in\left]0,1\right]$ and if the empirical spectral measure of $A^{}_NA_N^{*}$ converges weakly to some compactly supported probability distribution $\nu\neq\delta_0$, Dozier and Silverstein established in [R.B. Dozier and J.W. Silverstein, \textit{On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices}, J. Multivariate Anal. \textbf{98} (2007), no. 4, 678--694] that almost surely the empirical spectral measure of $M_N$ converges weakly towards a nonrandom distribution $\mu_{\sigma,\nu,c}$. In [Z.D. Bai and J.W. Silverstein, \textit{No eigenvalues outside the support of the limiting spectral distribution of information-plus-noise type matrices}, Random Matrices Theory Appl. \textbf{1} (2012), no. 1, 1150004, 44], Bai and Silverstein proved, under certain assumptions on the model, that for some closed interval in $\left]0;+\infty\right[$ outside the support of $\mu_{\sigma,\nu,c}$ satisfying some conditions involving $A_N$, almost surely no eigenvalues of $M_N$ will appear in this interval for all $N$ large. In this paper, we carry on studying the support of the limiting spectral measure previously investigated in [R.B. Dozier and J.W. Silverstein, \textit{Analysis of the limiting spectral distribution of large dimensional information-plus-noise type matrices}, J. Multivariate Anal. \textbf{98} (2007), no. 6, 1099--1122] and later in [Ph. Loubaton and P. Vallet, \textit{Almost sure localization of the eigenvalues in a Gaussian information plus noise model---application to the spiked models}, Electron. J. Probab. \textbf{16} (2011), no. 70, 1934--1959], [P. Vallet, Ph. Loubaton, and X. Mestre, \textit{Improved subspace estimation for multivariate observations of high dimension: the deterministic signals case}, IEEE Trans. Inform. Theory \textbf{58} (2012), no. 2, 1043--1068], and we show that, under almost the same assumptions as in [Z.D. Bai and J.W. Silverstein, \textit{No eigenvalues outside the support of the limiting spectral distribution of information-plus-noise type matrices}, Random Matrices Theory Appl. \textbf{1} (2012), no. 1, 1150004, 44], there is an exact separation phenomenon between the spectrum of $M_N$ and the spectrum of $A^{}_NA_N^{*}$: to a gap in the spectrum of $M_N$ pointed out by Bai and Silverstein there corresponds a gap in the spectrum of $A^{}_NA_N^{*}$ that splits the spectrum of $A^{}_NA_N^{*}$ exactly as that of $M_N$. We use the previous results to characterize the outliers of spiked Information-plus-noise type models.