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

A. E. Holroyd ; Y. Peres ; Jeffrey Steif (Institutionen för matematiska vetenskaper, matematik)
Electronic Communications in Probability (1083-589X). Vol. 19 (2014), p. 1-9.
[Artikel, refereegranskad vetenskaplig]

Let X-1 , X-2 , ... be independent identically distributed nonnegative random variables. Wald's identity states that the random sum S-T := X-1 + ... + X-T has expectation ET . EX1 provided T is a stopping time. We prove here that for any 1 < alpha <= 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 is given in terms of a prophet being more rewarded than a gambler when a certain impatience restriction is imposed.

Nyckelord: Wald's identity, stopping time, moment condition, prophet inequality

Denna post skapades 2015-01-16. Senast ändrad 2015-01-16.
CPL Pubid: 210882


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