Wald for non-stopping times: The rewards of impatient prophets

Let $X_1,X_2,\dots$ be independent identically distributed nonnegative random variables. Wald’s identity states that the random sum $S_T:=X_1+\cdots +X_T$ has expectation $E(T))E(X_1)$ provided $T$ is a stopping time. We prove here that for any $1<\alpha \le 2$, 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 $1\mathrm{/}(\alpha -1)$-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 of this is given in terms of a prophet being more rewarded than a gambler when a certain impatience restriction is imposed.