In the mathematical theory of probability, a Doob martingale (named after Joseph L. Doob,[1] also known as a Levy martingale) is a stochastic process that 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

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 function   satisfies the bounded differences property if substituting the value of the  th coordinate   changes the value of   by at most  . More formally, if there are constants   such that for all  , and all  ,

 

McDiarmid's Inequality[1] — Let   satisfy the bounded differences property with bounds  .

Consider independent random variables   where   for all  . Then, for any  ,

 
 

and as an immediate consequence,

 

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. Vol. 101. New York: Wiley. p. 293.