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.