On tails of stationary measures on a class of solvable groups

Abstract

Let G be a subgroup of GL ( R , d ) and let ( Q n , M n ) be a sequence of i.i.d. random variables with values in R d ⋊ G and law μ. Under some natural conditions there exists a unique stationary measure ν on R d of the process X n = M n X n − 1 + Q n . Its tail properties, i.e. behavior of ν { x : | x | > t } as t tends to infinity, were described some over thirty years ago by H. Kesten, whose results were recently improved by B. de Saporta, Y. Guivarc'h and E. Le Page. In the present paper we study the tail of ν in the situation when the group G 0 is Abelian and R d is replaced by a more general nilpotent Lie group N. Thus the tail behavior of ν is described for a class of solvable groups of type NA, i.e. being semi-direct extension of a simply connected nilpotent Lie group N by an Abelian group isomorphic to R d . Then, due to A. Raugi, ( N , ν ) can be interpreted as the Poisson boundary of ( N A , μ ).