Doob martingale

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A Doob martingale (named after Joseph L. Doob,[1] also known as a Levy martingale) is a mathematical construction of a stochastic process 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.[clarification needed]

Definition[edit]

Let be any random variable with . Suppose is a filtration, i.e. when . Define

then is a martingale,[2] namely Doob martingale, with respect to filtration .

To see this, note that

  • ;
  • as .

In particular, for any sequence of random variables on probability space and function such that , one could choose

and filtration such that

i.e. -algebra generated by . Then, by definition of Doob martingale, process where

forms a Doob martingale. Note that . This martingale can be used to prove McDiarmid's inequality.

McDiarmid's inequality[edit]

Statement[1][edit]

Consider independent random variables on probability space where for all and a mapping . Assume there exist constant such that for all ,

(In other words, changing the value of the th coordinate changes the value of by at most .) Then, for any ,

and

Proof[edit]

Pick any such that the value of is bounded, then, for any , by triangle inequality,

thus is bounded.

Define for all and . Note that . Since is bounded, by the definition of Doob martingale, forms a martingale. Now define

Note that and are both -measurable. In addition,

where the third equality holds due to the independence of . Then, applying the general form of Azuma's inequality to , we have

One-sided bound from the other direction is obtained by applying Azuma's inequality to and two-sided bound follows from union bound.

See also[edit]

References[edit]

  1. ^ a b Doob, J. L. (1940). "Regularity properties of certain families of chance variables" (PDF). Transactions of the American Mathematical Society. 47 (3): 455–486. doi:10.2307/1989964. JSTOR 1989964.
  2. ^ Doob, J. L. (1953). Stochastic processes. 101. New York: Wiley. p. 293.