On Kesten's Multivariate Choquet-Deny Lemma

2013/02/21 by Sebastian Mentemeier, Mentemeier, Sebastian
Mathematics · #60B15 Secondary: 46A55 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Primary: 60K15 #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #math.PR #msc:46A55 #msc:60B15 #msc:60K15

paper · pdf · doi:10.48550/arxiv.1302.5284

Second, corrected version

openalex publication_date 2013/02/21 · arxiv created 2014/03/14 · arxiv updated 2014/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let d >1 and (A_n)_{n ≥ 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.

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