An ergodic theorem for subadditive random functions on vector semigroups

Abstract

Let $f=(f^x\mid x\in S)$, $S\subset {\mathbb{Z}}^m$, be a semigroup of ergodic measure-preserving transformations of a probability space $(\Omega ,\mathsf{P})$ and $h$ a real random function on $S$, such that $h(x+y,\omega )\le h(x,\omega )+h(y,f^x\omega )$ for all $x,y\in S$ and $\omega \in \Omega$. We prove that there exists a sublinear function $q:O\to [-\infty ;\infty )$ defined on $O=\mathrm{int}(\mathrm{cone}(S))$, and a set $W\subset \Omega$ of full probability, such that $h(x_n,\omega )/|x_n|\to q(x)$ for all $\omega \in W$ and all sequences $(x_n)\subset S$ with asymptotic direction $x\in O$. The moment condition for this reflects the size of the semigroup $f$, not that of $S$. However, an additional independence assumption about $h$ is made.