On the universality of spectral limit for random matrices with martingale differences entries

TL;DR

This paper demonstrates that the spectral distribution of certain symmetric random matrices with martingale difference entries converges to classical laws like the semicircle or Marchenko-Pastur, extending known results from i.i.d. to dependent cases.

Contribution

It establishes spectral convergence results for matrices with martingale difference entries, generalizing classical laws to dependent structures under mild conditions.

Findings

Spectral distribution converges to semicircle law under regularity conditions.

Marchenko-Pastur law also holds for stationary martingale difference matrices.

Results apply to complex models including nonlinear ARCH fields.

Abstract

For a class of symmetric random matrices whose entries are martingale differences adapted to an increasing filtration, we prove that under a Lindeberg-like condition, the empirical spectral distribution behaves asymptotically similarly to a corresponding matrix with independent centered Gaussian entries having the same variances. Under a slightly reinforced condition, the approximation holds in the almost sure sense. We also point out several sufficient regularity conditions imposed to the variance structure for convergence to the semicircle law or the Marchenko-Pastur law and other convergence results. In the stationary case we obtain a full extension from the i.i.d. case to the martingale case of the convergence to the semicircle law as well as to the Marchenko-Pastur one. Our results are well adapted to study several examples including non linear ARCH infinite random fields.