An ergodic theorem for subadditive random functions on vector semigroups

TL;DR

This paper establishes an ergodic theorem for subadditive random functions on vector semigroups, showing convergence to a sublinear limit function under certain conditions, extending classical results to more general algebraic structures.

Contribution

It introduces a new ergodic theorem for subadditive functions on vector semigroups, with a focus on the size of the semigroup and additional independence assumptions.

Findings

Existence of a sublinear limit function q on the cone of the semigroup.

Almost sure convergence of normalized functions h(x,ω)/|x| to q(x).

The moment condition depends on the size of the semigroup, not the entire set S.

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 $q\colon O\to[-\infty;\infty)$ defined on $O=\mathrm{int}(\mathrm{cone}(S))$, and a set $W\subset\Omega$ of full probability, such that $h(x_n,\omega)/\lvert x_n\rvert\to q(x)$ 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.