A Conditioned Local Limit Theorem for Nonnegative Random Matrices

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.