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