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 [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)