Wald for non-stopping times: The rewards of impatient prophets
Journal article, 2014

Let X-1 , X-2 , ... be independent identically distributed nonnegative random variables. Wald's identity states that the random sum S-T := X-1 + ... + X-T has expectation ET . EX1 provided T is a stopping time. We prove here that for any 1 < alpha <= 2, if T is an arbitrary nonnegative random variable, then S-T has finite expectation provided that X-1 has finite alpha-moment and T has finite 1/(alpha - 1)-moment. We also prove a variant in which T is assumed to have a finite exponential moment. These moment conditions are sharp in the sense that for any i.i.d. sequence X-i violating them, there is a T satisfying the given condition for which S-T (and, in fact, X-T) has infinite expectation. An interpretation is given in terms of a prophet being more rewarded than a gambler when a certain impatience restriction is imposed.

Wald's identity

prophet inequality

moment condition

stopping time


A. E. Holroyd

Y. Peres

Jeffrey Steif

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

Electronic Communications in Probability

1083-589X (ISSN)

Vol. 19 1-9

Subject Categories




More information