A Glivenko-Cantelli Theorem for almost additive functions

Abstract

We develop a Glivenko--Cantelli theory for monotone, almost additive functions of i.\,i.\,d.\ sequences of random variables indexed by~${\mathbb{Z}}^d$. Under certain conditions on the random sequence, short range correlations are allowed as well. We have an explicit error estimate, consisting of a probabilistic and a geometric part. We apply the results to yield uniform convergence for several quantities arising naturally in statistical physics.

31 pages, to appear in Stochastic Processes and Applications

References in corpus (1)

• Spectral properties of Anderson-percolation Hamiltonians

Cited by in corpus (1)

• Glivenko-Cantelli Theory, Ornstein-Weiss quasi-tilings, and uniform Ergodic Theorems for distribution-valued fields over amenable groups