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In mathematics, Doob's martingale inequality, also known as Kolmogorov’s submartingale inequality is a result in the study of stochastic processes.It gives a bound on the probability that a submartingale exceeds any given value over a given interval of time.
A convex function of a martingale is a submartingale, by Jensen's inequality. For example, the square of the gambler's fortune in the fair coin game is a submartingale (which also follows from the fact that X n 2 − n is a martingale).
The Doob martingale was introduced by Joseph L. Doob in 1940 to establish concentration inequalities such as McDiarmid's inequality, which applies to functions that satisfy a bounded differences property (defined below) when they are evaluated on random independent function arguments.
A similar inequality was proved under weaker assumptions by Sergei Bernstein in 1937. Hoeffding proved this result for independent variables rather than martingale differences, and also observed that slight modifications of his argument establish the result for martingale differences (see page 9 of his 1963 paper).
The following result, called Doob's upcrossing inequality or, sometimes, Doob's upcrossing lemma, is used in proving Doob's martingale convergence theorems. [3] A "gambling" argument shows that for uniformly bounded supermartingales, the number of upcrossings is bounded; the upcrossing lemma generalizes this argument to supermartingales with ...
The following argument employs discrete martingales.As argued in the discussion of Doob's martingale inequality, the sequence ,, …, is a martingale. Define () = as follows. . Let =, a
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The following proof of McDiarmid's inequality [2] constructs the Doob martingale tracking the conditional expected value of the function as more and more of its arguments are sampled and conditioned on, and then applies a martingale concentration inequality (Azuma's inequality).