Recurrent Random Walks and Invariant Measures on Semigroups of n x n Matrices.
Let N be a simply connected nilpotent Lie group and let be a semidirect product, acting on N by diagonal automorphisms. Let (Qₙ,Mₙ) be a sequence of i.i.d. random variables with values in S. Under natural conditions, including contractivity in the mean, there is a unique stationary measure ν on N for the Markov process Xₙ = MₙXn-1 + Qₙ. We prove that for an appropriate homogeneous norm on N there is χ₀ such that . In particular, this applies to classical Poisson kernels on symmetric spaces,...