ASYMPTOTIC EXPANSIONS IN CENTRAL AND LOCAL LIMIT THEOREMS FOR PRODUCTS OF RANDOM MATRICES

Thi Thuy Bui (1) , Ion Grama (1) , Quansheng Liu (1)

1 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique

Let $ (A_n)_{n \geq 1} $ be a sequence of independent and identically distributed random $d \times d$ real matrices. Set $ G_n = A_n \ldots A_1 $, $ X_n^x = \dfrac{G_n x }{\vert G_n x\vert}$ and $S_n^x: =\log \vert G_nx\vert.$ We consider asymptotic properties of the Markov chain $(X_n^x , S_n^x) $. For invertible matrices, Le Page (1982) established a central limit theorem and a local limit theorem on $(X_n^x , S_n^x) $ with $ x $ a starting point on the unit sphere in $ \mathbb R^d $. In this paper, motivated by some applications in branching random walks, we improve and extend his theorems in the sense that: 1) we prove that the central limit theorem holds uniformly in $x$, and give an asymptotic expansion in the local limit theorem with a continuous function $f$ acting on $X_n^x$ and a directly Riemann integrable function $h$ acting on $S_n^x$; 2) we extend the results to the case of nonnegative matrices. Our approach is mainly based on the spectral gap theory recently developed for products of random matrices, and smoothing techniques for the approximation of functions.

Mots clés: Products of random matrices; Markov walk; central limit theorem; local limit theorem; asymptotic expansion

Domaines: Probabilités [math.PR]

Thi Thuy Bui, Ion Grama, Quansheng Liu. ASYMPTOTIC EXPANSIONS IN CENTRAL AND LOCAL LIMIT THEOREMS FOR PRODUCTS OF RANDOM MATRICES. 2020. ⟨hal-02910314⟩