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Regular behavior at infinity of stationary measures of stochastic recursion on NA groups

Dariusz Buraczewski, Ewa Damek (2010)

Colloquium Mathematicae

Let N be a simply connected nilpotent Lie group and let S = N ( ) d be a semidirect product, ( ) d 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 l i m t t χ ν x : | x | > t = C > 0 . In particular, this applies to classical Poisson kernels on symmetric spaces,...

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