## A note on Kesten’s Choquet-Deny lemma.(English)Zbl 1333.60159

Electron. Commun. Probab. 18, Paper No. 65, 7 p. (2013); erratum ibid. 19, Paper No. 20, 2 p. (2014).
Summary: 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 $$\mathbb{R}^d_{\geq} \setminus \{0\} = \mathbb{S}_\geq \times \mathbb{R}_>$$. We study harmonic functions of this Markov chain. In particular, it is shown that all bounded harmonic functions in $$\mathcal{C}_b(\mathbb{S}_\geq) \otimes\mathcal{C}_b(\mathbb{R}_>)$$ are constant. The idea of the proof is originally due to H. Kesten [Ann. Probab. 2, 355–386 (1974; Zbl 0303.60090)], but is considerably shortened here. A similar result for invertible matrices is given as well.

### MSC:

 60J05 Discrete-time Markov processes on general state spaces 60B20 Random matrices (probabilistic aspects) 60K15 Markov renewal processes, semi-Markov processes 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 46A55 Convex sets in topological linear spaces; Choquet theory

### Keywords:

Markov random walks; random matrices; Choquet-Deny lemma

Zbl 0303.60090
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