Doob martingale

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A Doob martingale (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]

A Doob martingale (named after Joseph L. Doob)[1] is a generic construction that is always a martingale. Specifically, consider any set of random variables

taking values in a set for which we are interested in the function and define:

where the above expectation is itself a random quantity since the expectation is only taken over

and

are treated as random variables. It is possible to show that is always a martingale regardless of the properties of .[citation needed]

The sequence is the Doob martigale for f.[2]

Application[edit]

Thus if one can bound the differences

,

one can apply Azuma's inequality and show that with high probability is concentrated around its expected value

McDiarmid's inequality[edit]

One common way of bounding the differences and applying Azuma's inequality to a Doob martingale is called McDiarmid's inequality.[3]

Suppose are independent and assume that satisfies

(In other words, replacing the -th coordinate by some other value changes the value of by at most .)

It follows that

and therefore Azuma's inequality yields the following McDiarmid inequalities for any :

and

and

See also[edit]

Notes[edit]

  1. ^ 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. 
  2. ^ Anupam Gupta (2011) http://www.cs.cmu.edu/~avrim/Randalgs11/lectures/lect0321.pdf Lecture notes
  3. ^ McDiarmid, Colin (1989). "On the Method of Bounded Differences". Surveys in Combinatorics. 141: 148–188. 

References[edit]