Math 262a Class Notes, 4-13-04

so that this process is a martingale. More often that not, martingales are used in combinatorics to show that a particular random variable is concentrated around its mean. We can show (via the second moment method, for example) that for the above sequence X0, . . . , Xt describing a t-step random walk, the outcome of Xt is concentrated in an interval of length c √ t. This concentration property is not unique to the random walk, and in fact it is shared by all martingales satisfying the Lipschitz condition |Xi+1 −Xi| ≤ 1 for all 0 ≤ i ≤ t− 1.