Limiting spectral distribution of Gram matrices associated with functionals of $\beta$-mixing processes

Abstract

We give asymptotic spectral results for Gram matrices of the form $ n^{-1}\mathcal{X}_n \mathcal{X}_n^T$ where the entries of $\mathcal{X}_n$ are dependent across both rows and columns. More precisely, they consist of short or long range dependent random variables having moments of second order and that are functionals of an absolutely regular sequence. We also give a concentration inequality of the Stieltjes transform and we prove that, under an arithmetical decay condition on the $\beta$-mixing coefficients, it is almost surely concentrated around its expectation. Applications to examples of positive recurrent Markov chains and dynamical systems are also given.