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

• Published in 2021
In the collection
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.

### BibTeX entry

@article{AGamblerthatBetsForeverandtheStrongLawofLargeNumbers,
title = {A Gambler that Bets Forever and the Strong Law of Large Numbers},
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.},
url = {http://arxiv.org/abs/2105.03803v1 http://arxiv.org/pdf/2105.03803v1},
year = 2021,
author = {Calvin Wooyoung Chin},
comment = {},
urldate = {2021-06-07},
archivePrefix = {arXiv},
eprint = {2105.03803},
primaryClass = {math.PR},
collections = {probability-and-statistics}
}