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.

MSC::

• Primary 60G50
• Secondary 60F15

Additional information

Funding

National Research Foundation of Korea, ID 6524, Grant Nos. N01210444 and N01210362