# Martingales

• Jiming Jiang
Chapter
Part of the Springer Texts in Statistics book series (STS, volume 0)

## Abstract

The term martingale originally referred to a betting strategy. Imagine a gambler playing a blackjack game (also known as twenty-one) in a casino (if you have not been in a casino or have never heard about the blackjack, there is nothing to worry, as far as this book is concerned). He begins with an initial bet of \$5, which is the minimal according to the rule of the casino table. Every time he loses, he doubles the bet; otherwise he returns to the minimal bet. For example, a sequence of bettings may be \$5 (lose), \$10 (lose), \$20 (lose), \$40 (lose), \$80 (win), \$5 (lose), \$10 (lose), .... It is easy to see that with this strategy, as long as the gambler does not keep losing, whenever he wins he recovers all his previous losses, plus an additional \$5, which is equal to his initial bet (Exercise 8.1). However, \$5 is as much as he can win at the end of any losing sequence, and he is risking more and more in order to win the \$5 as the sequence extends longer and longer.

## Keywords

Independent Random Variable Invariance Principle Strong Limit Theorem Martingale Convergence Theorem REML Estimator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.