A Glivenko-Cantelli Theorem for almost additive functions

TL;DR

This paper establishes a Glivenko-Cantelli type theorem for almost additive functions of i.i.d. sequences, allowing for short-range correlations, with explicit error estimates, and applies these results to problems in statistical physics.

Contribution

It extends Glivenko-Cantelli theory to almost additive functions of dependent random sequences with explicit error bounds.

Findings

Proves uniform convergence for almost additive functions under certain conditions.

Provides explicit probabilistic and geometric error estimates.

Applies the theory to key quantities in statistical physics.

Abstract

We develop a Glivenko--Cantelli theory for monotone, almost additive functions of i.\,i.\,d.\ sequences of random variables indexed by~$\Z^d$. 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.