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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

Citations:

Zbl 0303.60090
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