A Conditioned Local Limit Theorem for Nonnegative Random Matrices

For any fixed real a > 0 and x ∈ℝ^d, d ≥ 1 , we consider the real-valued random process (S_n)_n ≥ 0 defined by S_0= a, S_n= a+ln| g_n⋯ g_1x| , n ≥ 1 , where the g_k, k ≥ 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 ≥ 0 remains nonnegative up to time n and simultaneously belongs to some compact set [b, b+ℓ ]⊂ℝ^+_* at time n.