Discrete Time Martingales and Concentration Inequalities
For an independent sequence of random variables X 1, X 2, …, the conditional expectation of the present term of the sequence given the past terms is the same as its unconditional expectation. Martingales let the conditional expectation depend on the past terms, but in a special way. Thus, similar to Markov chains, martingales act as natural models for incorporating dependence into a sequence of observed data. But the value of the theory of martingales is much more than simply its modeling value. Martingales arise, as natural byproducts of the mathematical analysis in an amazing variety of problems in probability and statistics. Therefore, results from martingale theory can be immediately applied to all these situations in order to make deep and useful conclusions about numerous problems in probability and statistics. A particular modern set of applications of martingale methods is in the area of concentration inequalities, which place explicit bounds on probabilities of large deviations of functions of a set of variables from their mean values. This chapter gives a glimpse into some important concentration inequalities, and explains how martingale theory enters there. Martingales form a nearly indispensable tool for probabilists and statisticians alike.
KeywordsMaximal Inequality Sequential Probability Ratio Test Concentration Inequality Martingale Theory Fair Game
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