Wald identity

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An identity in sequential analysis which states that the mathematical expectation of the sum of a random number of independent, identically-distributed random variables is equal to the product of the mathematical expectations and :

A sufficient condition for the Wald identity to be valid is that the mathematical expectations and in fact exist, and for the random variable to be a Markov time (i.e. for any the event is determined by the values of the random variables or, which is the same thing, the event belongs to the -algebra generated by the random variables ). Wald's identity is a particular case of a fundamental theorem in sequential analysis stating that


for all complex for which exists and . It was established by A. Wald [1].


[1] A. Wald, "Sequential analysis", Wiley (1952)
[2] W. Feller, "An introduction to probability theory and its applications", 1, Wiley (1957) pp. Chapt.14


The general result (*) is (also) referred to as Wald's formula.


[a1] A.V. [A.V. Skorokhod] Skorohod, "Random processes with independent increments" , Kluwer (1991) pp. 23 (Translated from Russian)
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Wald identity. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article