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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.
KeywordsIndependent Random Variable Invariance Principle Strong Limit Theorem Martingale Convergence Theorem REML Estimator
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