Simple Example of Azuma's Inequality For Coin Flips
Let Fi be a sequence of independent and identically distributed random coin flips (i.e., let Fi be equally likely to be -1 or 1 independent of the other values of Fi). Defining yields a martingale with |Xk − Xk−1| ≤ 1, allowing us to apply Azuma's inequality. Specifically, we get
For example, if we set t proportional to N, then this tells us that although the maximum possible value of XN scales linearly with N, the probability that the sum scales linearly with N decreases exponentially fast with N.
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