Products of random mappings

Abstract

Suppose that $X^1,X^2,\dotsc$ is a stationary stochastic process of positive $k\times k$ matrices, and let ${}^nY^1=X^nX^{n-1}\dots X^1$ be the corresponding product matrices. For a special case, Bellman showed that the elements $[{}^nY^1]_{ij}$ converge in the sense that $n^{-1}\mathrm{E}\{\log[{}^nY^1]_{ij}\}\rightarrow a$ as $n\rightarrow\infty$. The constant $a$ is independent of $i$ and $j$. Bellman also conjectured that, asymptotically, the $n^{-1/2}\{\log[{}^nY^1]_{ij}-na\}$ terms are distributed according to a normal distribution with a common variance, independent of $ij$. Later Furstenberg and Kesten generalized and strengthened Bellman's result and established the validity of his conjecture.

This paper extends these results to the case of nonlinear mappings that are monotonic and homogeneous of degree one on $R^k_+$. Specifically, given a stationary process $H^1,H^2,\dots$ of such mappings, we define the composite mappings ${}^nF^1(\cdot)=H^n(H^{n-1}(\dots (H^1(\cdot))\dots)$. Under appropriate conditions, the components $[{}^nF^1(x^0)]_i$ have the property that, almost surely, $n^{-1}\log[{}^nF^1(x^0)]_i\rightarrow a$ independent of $x^0$ and $i$. Furthermore the components $n^{-1/2}\{\log[{}^nF^1(x^0)]_i-na\}$ are asymptotically distributed according to a normal distribution with a common variance.