Electronic Communications in Probability

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

Alexander Holroyd, Yuval Peres, and Jeffrey Steif

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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 $\mathbb{E} T \cdot \mathbb{E} X_1$ 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.

Article information

Electron. Commun. Probab., Volume 19 (2014), paper no. 78, 9 pp.

Accepted: 12 November 2014
First available in Project Euclid: 7 June 2016

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Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks

Wald's identity stopping time moment condition prophet inequality

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Holroyd, Alexander; Peres, Yuval; Steif, Jeffrey. Wald for non-stopping times: the rewards of impatient prophets. Electron. Commun. Probab. 19 (2014), paper no. 78, 9 pp. doi:10.1214/ECP.v19-3609. https://projecteuclid.org/euclid.ecp/1465316780

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