## Electronic Communications in Probability

### A note on Kesten's Choquet-Deny lemma

Sebastian Mentemeier

#### 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

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

Rights

#### 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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