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» Encyclopaedia of Mathematics » W » Wald identity

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Wald identity

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].

References

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

A.N. Shiryaev

Comments

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

References

[a1] | A.V. [A.V. Skorokhod] Skorohod,   "Random processes with independent increments" , Kluwer  (1991)  pp. 23  (Translated from Russian)

This text originally appeared in Encyclopaedia of Mathematics - ISBN 1402006098

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