A Gambler that Bets Forever and the Strong Law of Large Numbers

TL;DR

This paper presents a simple proof that a gambler with a positive expected value will not go broke with positive probability, and uses this to give an elementary proof of the strong law of large numbers, connecting probability theory and ergodic theorems.

Contribution

It introduces an elementary proof linking gambler's ruin probabilities with the strong law of large numbers, using ideas from ergodic theory.

Findings

Gambler with positive expected value does not go broke with positive probability

Elementary proof of the strong law of large numbers

Connection between gambler's ruin and ergodic theorems

Abstract

In this expository note, we give a simple proof that a gambler repeating a game with positive expected value never goes broke with a positive probability. This does not immediately follow from the strong law of large numbers or other basic facts on random walks. Using this result, we provide an elementary proof of the strong law of large numbers. The ideas of the proofs come from the maximal ergodic theorem and Birkhoff's ergodic theorem.