Wald for Non-Stopping Times: The Rewards of Impatient Prophets

Let X₁, X₂, . . . be independent identically distributed nonnegative random variables. Wald’s identity states that the random sum S_T := X₁ + · · · + X_T has expectation E_T · EX₁ provided T is a stopping time. We prove here that for any 1 < α ≤ 2, if T is an arbitrary nonnegative random variable, then S_T has finite expectation provided that X₁ has finite α-moment and T has finite 1/(α − 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 of this is given in terms of a prophet being more rewarded than a gambler when a certain impatience restriction is imposed.