Universal Gaussian fluctuations of non-Hermitian matrix ensembles: from weak convergence to almost sure CLTs

Ivan Nourdin (1) , Giovanni Peccati (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 Unité de Recherche en Mathématiques

In the paper [25], written in collaboration with Gesine Reinert, we proved a universality principle for the Gaussian Wiener chaos. In the present work, we aim at providing an original example of application of this principle in the framework of random matrix theory. More specifically, by combining the result in [25] with some combinatorial estimates, we are able to prove multi-dimensional central limit theorems for the spectral moments (of arbitrary degrees) associated with random matrices with real-valued i.i.d. entries, satisfying some appropriate moment conditions. Our approach has the advantage of yielding, without extra effort, bounds over classes of smooth (i.e., thrice differentiable) functions, and it allows to deal directly with discrete distributions. As a further application of our estimates, we provide a new ``almost sure central limit theorem'', involving logarithmic means of functions of vectors of traces.

Mots clés: Almost sure central limit theorems; Central limit theorems; Eigenvalues; Fourth-moment criteria; Invariance principles; Non-Hermitian random matrices; Normal approximation; Spectral moments; Universality

Domaines: Probabilités [math.PR]

Ivan Nourdin, Giovanni Peccati. Universal Gaussian fluctuations of non-Hermitian matrix ensembles: from weak convergence to almost sure CLTs. 2010. ⟨hal-00453622⟩