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2013 A note on Kesten's Choquet-Deny lemma
Sebastian Mentemeier
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Electron. Commun. Probab. 18: 1-7 (2013). DOI: 10.1214/ECP.v18-2629

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)

Citation

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Sebastian Mentemeier. "A note on Kesten's Choquet-Deny lemma." Electron. Commun. Probab. 18 1 - 7, 2013. https://doi.org/10.1214/ECP.v18-2629

Information

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

zbMATH: 1333.60159
MathSciNet: MR3091723
Digital Object Identifier: 10.1214/ECP.v18-2629

Subjects:
Primary: 60K15
Secondary: 46A55, 60B15

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