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A note on Kesten's Choquet-Deny lemma

   
@article{ECP2629,
	author = {Sebastian Mentemeier},
	title = {A note on Kesten's Choquet-Deny lemma},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {18},
	year = {2013},
	keywords = {Choquet-Deny Lemma, Markov Random Walks, Products of Random Matrices},
	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).
},
	pages = {no. 65, 1-7},
	issn = {1083-589X},
	doi = {10.1214/ECP.v18-2629},    
        url = {http://ecp.ejpecp.org/article/view/2629}}