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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]

DefinitionEdit

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 inequalityEdit

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

 

ProofEdit

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 alsoEdit

ReferencesEdit

  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.