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A note on Kesten's Choquet-Deny lemma


 
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1. Title Title of document A note on Kesten's Choquet-Deny lemma
 
2. Creator Author's name, affiliation, country Sebastian Mentemeier; University of Münster; Germany
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Choquet-Deny Lemma, Markov Random Walks, Products of Random Matrices
 
3. Subject Subject classification 60K15; 60B15; 46A55
 
4. Description 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).
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Deutsche Forschungsgemeinschaft (SFB 878)
 
7. Date (YYYY-MM-DD) 2013-08-05
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/2629
 
10. Identifier Digital Object Identifier 10.1214/ECP.v18-2629
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 18
 
12. Language English=en en
 
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