General Glivenko-Cantelli theorems

Abstract

A Glivenko–Cantelli theorem is a fundamental result in statistics. It says that an empirical distribution function uniformly approximates the true distribution function for a sufficiently large sample size. We prove general Glivenko–Cantelli theorems for three types of sequences of random variables: delayed regenerative, delayed stationary and delayed exchangeable. In particular, our results hold for irreducible Harris recurrent Markov chains that admit a stationary probability distribution but are not necessarily in the stationary state. We also do not assume any mixing conditions on the Markov chain. This is useful in the application of Markov chain Monte Carlo methods. A key tool used is a generalized version of Polya's theorem on the convergence of distribution functions.