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

TL;DR

This paper investigates the spectral separation in large Information-Plus-Noise matrices, establishing an exact correspondence between gaps in the spectra of the matrices and their deterministic parts, with applications to spiked models.

Contribution

It proves an exact separation phenomenon for eigenvalues, linking spectral gaps of large random matrices to those of their deterministic components, extending prior results.

Findings

Spectral gaps in $M_N$ correspond exactly to gaps in $A_NA_N^*$.

Characterization of outliers in spiked models.

Extension of previous spectral support results.

Abstract

We consider large Information-Plus-Noise type matrices of the form $M_N=(\sigma \frac{X_N}{\sqrt{N}}+A_N)(\sigma \frac{X_N}{\sqrt{N}}+A_N)^*$ 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 $n/N \rightarrow c\in ]0,1]$ and if the empirical spectral measure of $A_N A_N^*$ converges weakly to some compactly supported probability distribution $\nu \neq \delta_0$, Dozier and Silverstein established that almost surely the empirical spectral measure of $M_N$ converges weakly towards a nonrandom distribution $\mu_{\sigma,\nu,c}$. Bai and Silverstein proved, under certain assumptions on the model, that for some closed interval in $]0;+\infty[$ 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 with the study of the support of the limiting spectral measure previously investigated by Dozier and Silverstein and later by Vallet, Loubaton and Mestre and Loubaton and P. Vallet, and we show that, under almost the same assumptions as Bai and Silvertein, 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, it corresponds a gap in the spectrum of $A_NA_N^*$ which 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.