Namespaces
Variants
Actions

Wald identity

From Encyclopedia of Mathematics
Revision as of 17:25, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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


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)
How to Cite This Entry:
Wald identity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wald_identity&oldid=25963
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article