Let X1, X2, . . . be independent identically distributed nonnegative random variables. Wald’s identity states that the random sum ST := X1 + · · · + XT has expectation ET ·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 ST has finite expectation provided that X1 has finite α-moment and T has finite 1/(α− 1)-moment. ...
