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

TL;DR

This paper extends Wald's identity to non-stopping times by establishing sharp moment conditions under which the expected sum remains finite, highlighting the impact of impatience restrictions on prophet versus gambler rewards.

Contribution

It introduces new moment conditions ensuring finite expectation of sums with non-stopping times, generalizing Wald's identity beyond traditional stopping time assumptions.

Findings

Finite expectation of sums under non-stopping times depends on specific moment conditions.

Moment conditions are sharp; violations lead to infinite expectation.

Impatience restrictions influence the relative rewards of prophets and gamblers.

Abstract

Let $X_1,X_2,\ldots$ 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 $1<\alpha\leq 2$, 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 of this is given in terms of a prophet being more rewarded than a gambler when a certain impatience restriction is imposed.