# Wald identity

An identity in sequential analysis which states that the mathematical expectation of the sum $S _ \tau = X _ {1} + \dots + X _ \tau$ of a random number $\tau$ of independent, identically-distributed random variables $X _ {1} , X _ {2} \dots$ is equal to the product of the mathematical expectations ${\mathsf E} X _ {1}$ and ${\mathsf E} \tau$:

$${\mathsf E} ( X _ {1} + \dots + X _ \tau ) = \ {\mathsf E} X _ {1} \cdot {\mathsf E} \tau .$$

A sufficient condition for the Wald identity to be valid is that the mathematical expectations ${\mathsf E} | X _ {1} |$ and ${\mathsf E} \tau$ in fact exist, and for the random variable $\tau$ to be a Markov time (i.e. for any $n = 1, 2 \dots$ the event $\{ \tau = n \}$ is determined by the values of the random variables $X _ {1} \dots X _ {n}$ or, which is the same thing, the event $\{ \tau = n \}$ belongs to the $\sigma$- algebra generated by the random variables $X _ {1} \dots X _ {n}$). Wald's identity is a particular case of a fundamental theorem in sequential analysis stating that

$$\tag{* } {\mathsf E} \left [ e ^ {\lambda S _ \tau } ( \phi ( \lambda )) ^ {- \tau } \right ] = 1$$

for all complex $\lambda$ for which $\phi ( \lambda ) = {\mathsf E} e ^ {\lambda X _ {1} }$ exists and $| \phi ( \lambda ) | \geq 1$. It was established by A. Wald .

How to Cite This Entry:
Wald identity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wald_identity&oldid=49166
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article