Large Deviations of Products of Random Topical Operators

Abstract

A topical operator on $\mathbb{R}^d$ is one which is isotone and homogeneous. Let ${A(n) : n \geq 1}$ be a sequence of i.i.d. random topical operators such that the projective radius of $A(n) \dots A(1)$ is almost surely bounded for large $n$. If ${x(n) : n \geq 1}$, is a sequence of vectors given by $x(n) = A(n) \dots A(1)x_0$, for some fixed initial condition $x_0$, then the sequence ${x(n)/n : n \geq 1}$ satisfies a weak large deviation principle. As corollaries of this result we obtain large deviation principles for products of certain random aperiodic max-plus and min-plus matrix operators and for products of certain random aperiodic nonnegative matrix operators.