A note on Kesten's Choquet-Deny lemma

Let d > 1 and (A(n))(n) (is an element of) (N) be a sequence of independent identically distributed random d x d matrices with nonnegative entries. This induces a Markov chain M-n = A(n)M(n-1) on the cone R->=(d) \ {0} = S->= x R->. We study harmonic functions of this Markov chain. In particular, it is shown that all bounded harmonic functions in C-b (S->=) circle times C-b (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.