On Kesten's Multivariate Choquet-Deny Lemma

Abstract

Let $d>1$ and $(A_n{)}_{n\ge 1}$ be a sequence of independent identically distributed random matrices with nonnegative entries and no zero column. This induces a Markov chain $M_n=A_n{M}_{n-1}$ on the cone of d-vectors with nonnegative entries. We study harmonic functions of this Markov chain. Considering a polar decomposition $M_n=X_n\exp (S_n)$, where $X_n$ is a vector of unit length, and $S_n$ a real valued random variable, it is in particular shown that all "compound" harmonic functions $L(x,s)=f(x)g(s)$ are constant. The idea of the proof is originally due to Kesten [Renewal theory for functionals of a Markov chain with general state space, Ann. Prob. 2 (1974), 355 - 386], but is considerably shortened here. A similar result for invertible matrices is given as well.

Second, corrected version

References in corpus (2)

• On multidimensional Mandelbrot's cascades
• Spectral gap properties for linear random walks and Pareto's asymptotics for affine stochastic recursions