On Kesten's Multivariate Choquet-Deny Lemma

TL;DR

This paper investigates harmonic functions of a Markov chain induced by i.i.d. nonnegative matrices, proving that all compound harmonic functions are constant, extending Kesten's lemma to multivariate and invertible matrix cases.

Contribution

The paper extends Kesten's multivariate Choquet-Deny lemma to Markov chains generated by nonnegative and invertible matrices, showing all compound harmonic functions are constant.

Findings

All compound harmonic functions are constant.

Extension of Kesten's lemma to multivariate and invertible matrices.

Shortened proof of the original result.

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.