Related Inequalities
Doob's inequality for discrete-time martingales implies Kolmogorov's inequality: if X1, X2, ... is a sequence of real-valued independent random variables, each with mean zero, it is clear that
so Mn = X1 + ... + Xn is a martingale. Note that Jensen's inequality implies that is a nonnegative submartingale if is a martingale. Hence, taking p = 2 in Doob's martingale inequality,
which is precisely the statement of Kolmogorov's inequality.
Read more about this topic: Doob's Martingale Inequality
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