Let F_i be a sequence of independent and identically distributed random coin flips (i.e., let F_i be equally like to be +1 or −1 independent of the other values of F_i). Defining yields a martingale with , allowing us to apply Azuma's inequality. Specifically, we get A separate article treats the device for fastening horses bridles or dogs collars called a martingale. ... In probability theory, Azumas inequality gives a concentration result for the values of martingales that have bounded differences. ...

For example, if we set t proportional to N, then this tells us that although the maximum possible value of X_N scales linearly with N, the probability that the sum scales linearly with N decreases exponentially fast with N. A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ...