# A Note on the Birkhoff Ergodic Theorem

## TL;DR

This paper extends the Birkhoff ergodic theorem to Markov processes on Polish spaces with converging marginals, showing the theorem's conclusions hold in the $p$-th mean for bounded Lipschitz functions regardless of initial distribution.

## Contribution

It generalizes the classical Birkhoff ergodic theorem to non-invariant initial distributions under specific convergence conditions.

## Key findings

- The theorem holds in the $p$-th mean for bounded Lipschitz functions.
- Convergence in the $L^{1}$-Wasserstein metric is sufficient.
- Results apply to Markov processes on Polish spaces.

## Abstract

The classical Birkhoff ergodic theorem states that for an ergodic Markov process the limiting behaviour of the time average of a function (having finite $p$-th moment, $p\ge1$, with respect to the invariant measure) along the trajectories of the process, starting from the invariant measure, is a.s. and in the $p$-th mean constant and equals to the space average of the function with respect to the invariant measure. The crucial assumption here is that the process starts from the invariant measure, which is not always the case. In this paper, under the assumptions that the underlying process is a Markov process on Polish space, that it admits an invariant probability measure and that its marginal distributions converge to the invariant measure in the $L^{1}$-Wasserstein metric, we show that the assertion of the Birkhoff ergodic theorem holds in the $p$-th mean, $p\geq1$, for any bounded Lipschitz function and any initial distribution of the process.