Electronic Communications in Probability

A note on Kesten's Choquet-Deny lemma

Sebastian Mentemeier

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Let $d >1$ and $(A_n)_{n \in \mathbb{N}}$ be a sequence of independent identically distributed random matrices with nonnegative entries. This induces a Markov chain $M_n = A_n M_{n-1}$ on the cone $\mathbb{R}^d_{\ge} \setminus \{0\} = \mathbb{S}_\ge \times \mathbb{R}_>$. We study harmonic functions of this Markov chain. In particular, it is shown that all bounded harmonic functions in $\mathcal{C}_b(\mathbb{S}_\ge) \otimes\mathcal{C}_b(\mathbb{R}_>)$ 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.

There is an erratum in ECP volume 19 paper 20 (2014)

Article information

Electron. Commun. Probab., Volume 18 (2013), paper no. 65, 7 pp.

Accepted: 5 August 2013
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 46A55: Convex sets in topological linear spaces; Choquet theory [See also 52A07]

Choquet-Deny Lemma Markov Random Walks Products of Random Matrices

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Mentemeier, Sebastian. A note on Kesten's Choquet-Deny lemma. Electron. Commun. Probab. 18 (2013), paper no. 65, 7 pp. doi:10.1214/ECP.v18-2629. https://projecteuclid.org/euclid.ecp/1465315604

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