Electronic Communications in Probability

A note on Kesten's Choquet-Deny lemma

Sebastian Mentemeier

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465315604

Digital Object Identifier
doi:10.1214/ECP.v18-2629

Mathematical Reviews number (MathSciNet)
MR3091723

Zentralblatt MATH identifier
1333.60159

Subjects
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]

Keywords
Choquet-Deny Lemma Markov Random Walks Products of Random Matrices

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

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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