A Conditioned Local Limit Theorem for Nonnegative Random Matrices

Abstract

For any fixed real $a > 0$ and $x \in {\mathbb {R}}^d, d \ge 1$, we consider the real-valued random process $(S_n)_{n \ge 0}$ defined by $S_0= a, S_n= a+\ln \vert g_n\cdots g_1x\vert , n \ge 1$, where the $g_k, k \ge 1,$ are i.i.d. nonnegative random matrices. By using the strategy initiated by Denisov and Wachtel to control fluctuations in cones of d-dimensional random walks, we obtain an asymptotic estimate and bounds on the probability that the process $(S_n)_{n \ge 0}$ remains nonnegative up to time n and simultaneously belongs to some compact set $[b, b+\ell ]\subset {\mathbb {R}}^+_*$ at time n.