A Mass Transport Proof of the Ergodic Theorem

TL;DR

This paper uses the mass transport method to prove a generalized ergodic theorem, extending classical results about gambler's ruin to stationary gain sequences, and derives Birkhoff's ergodic theorem as a consequence.

Contribution

It introduces a mass transport proof for the ergodic theorem applicable to stationary gain sequences, broadening the scope beyond i.i.d. assumptions.

Findings

Proves a generalized gambler's ruin result for stationary sequences.

Derives Birkhoff's ergodic theorem from the mass transport approach.

Establishes a new proof technique for ergodic theory results.

Abstract

It is known that a gambler repeating a game with positive expected value has a positive probability to never go broke. We use the mass transport method to prove the generalization of this fact where the gains from the bets form a stationary, rather than an i.i.d., sequence. Birkhoff's ergodic theorem follows from this by a standard argument.