Martingales generalize the notion of a fair game in gambling. Theory to the contrary, many gamblers still believe that they simply need to hone their strategies to beat the house. Probabilists know better. The real payoff with martingales is their practical value throughout probability theory. This chapter introduces martingales, develops some relevant theory, and delves into a few applications. As a prelude, readers are urged to review the material on conditional expectations in Chapter 1. In the current chapter we briefly touch on the convergence properties of martingales, the optional stopping theorem, and large deviation bounds via Azuma’s inequality. More extensive treatments of martingale theory appear in the books [23, 24, 53, 80, 106, 118, 208]. Our other referenced sources either provide elementary accounts comparable in difficulty to the current material [129, 170] or interesting special applications [4, 134, 186, 201].
KeywordsIndependent Random Variable Fractional Linear Transformation Bernoulli Random Variable Longe Common Subsequence Martingale Property
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