A note on Kesten's Choquet-Deny lemma

Sebastian Mentemeier (University of Münster)


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).

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Pages: 1-7

Publication Date: August 5, 2013

DOI: 10.1214/ECP.v18-2629


  • Breiman, Leo. The strong law of large numbers for a class of Markov chains. Ann. Math. Statist. 31 1960 801--803. MR0117786
  • Breiman, Leo. Probability. Addison-Wesley Publishing Company, Reading, Mass.-London-Don Mills, Ont. 1968 ix+421 pp. MR0229267
  • Buraczewski, Dariusz; Damek, Ewa; Guivarc'h, Yves. On multidimensional Mandelbrot's cascades, arXiv:1109.1845.
  • Guivarc'h, Yves; Le Page, Émile. Spectral gap properties and asymptotics of stationary measures for affine random walks, arXiv:1204.6004.
  • Guivarc'h, Yves; Le Page, Émile. Simplicité de spectres de Lyapounov et propriété d'isolation spectrale pour une famille d'opérateurs de transfert sur l'espace projectif. (French) [Simplicity of the Lyapunov spectrum and spectral gap property for a family of transfer operators on projective space] Random walks and geometry, 181--259, Walter de Gruyter GmbH & Co. KG, Berlin, 2004. MR2087783
  • Guivarc'h, Yves; Le Page, Émile. Homogeneity at infinity of stationary solutions of multivariate affine stochastic recursions, Random Matrices and Iterated Random Functions: Münster, October 2011 (Matthias Löwe and Gerold Alsmeyer, eds.), Springer Proceedings in Mathematics & Statistics, vol. 53, Springer, 2013.
  • Kesten, Harry. Random difference equations and renewal theory for products of random matrices. Acta Math. 131 (1973), 207--248. MR0440724
  • Kesten, Harry. Renewal theory for functionals of a Markov chain with general state space. Ann. Probability 2 (1974), 355--386. MR0365740
  • Klüppelberg, Claudia; Pergamenchtchikov, Serguei. Renewal theory for functionals of a Markov chain with compact state space. Ann. Probab. 31 (2003), no. 4, 2270--2300. MR2016619
  • Sebastian Mentemeier. On Multivariate Stochastic Fixed Point Equations: The Smoothing Transform and Random Difference Equations, Ph.D. thesis, University of Münster, 2013.

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