  Let $d >1$ and $(A_n)_{n \ge 1}$ be a sequence of independent identically
distributed random matrices with nonnegative entries and no zero column. This
induces a Markov chain $M_n = A_n M_{n-1}$ on the cone of d-vectors with
nonnegative entries. We study harmonic functions of this Markov chain.
Considering a polar decomposition $M_n = X_n \exp(S_n)$, where $X_n$ is a
vector of unit length, and $S_n$ a real valued random variable, it is in
particular shown that all "compound" harmonic functions $L(x,s)=f(x)g(s)$ 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.
