A Conditioned Local Limit Theorem for Nonnegative Random Matrices

For any fixed real $$a > 0$$ a > 0 and $$x \in {\mathbb {R}}^d, d \ge 1$$ x ∈ R d , d ≥ 1 , we consider the real-valued random process $$(S_n)_{n \ge 0}$$ ( S n ) n ≥ 0 defined by $$ S_0= a, S_n= a+\ln \vert g_n\cdots g_1x\vert , n \ge 1$$ S 0 = a , S n = a + ln | g n ⋯ g 1 x | , n ≥ 1 , where the $$g_k, k \ge 1, $$ 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 $$(S_n)_{n \ge 0}$$ ( S n ) n ≥ 0 remains nonnegative up to time n and simultaneously belongs to some compact set $$[b, b+\ell ]\subset {\mathbb {R}}^+_*$$ [ b , b + ℓ ] ⊂ R ∗ + at time n.