Doob martingale

A Doob martingale (also known as a Levy martingale) is a mathematical construction of a which approximates a given random variable and has the martingale property with respect to the given filtration. It may be thought of as the evolving sequence of best approximations to the random variable based on information accumulated up to a certain time.

When analyzing sums, random walks, or other additive functions of independent random variables, one can often apply the central limit theorem, law of large numbers, Chernoff's inequality, Chebyshev's inequality or similar tools. When analyzing similar objects where the differences are not independent, the main tools are martingales and Azuma's inequality.

A Doob martingale (named after Joseph L. Doob) is a generic construction that is always a martingale. Specifically, consider any set of random variables

taking values in a set ${\displaystyle A}$ for which we are interested in the function ${\displaystyle f:A^{n}\to {\mathbb {R}}}$ and define:

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