{"id":339770,"date":"2016-12-20T17:02:48","date_gmt":"2016-12-21T01:02:48","guid":{"rendered":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=339770"},"modified":"2018-10-16T21:16:57","modified_gmt":"2018-10-17T04:16:57","slug":"wald-non-stopping-times-rewards-impatient-prophets","status":"publish","type":"msr-research-item","link":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/publication\/wald-non-stopping-times-rewards-impatient-prophets\/","title":{"rendered":"Wald for Non-Stopping Times: The Rewards of Impatient Prophets"},"content":{"rendered":"<p>Let <em>X<\/em><sub>1<\/sub>, <em>X<\/em><sub>2<\/sub>, . . . be independent identically distributed nonnegative random variables. Wald\u2019s identity states that the random sum <em>S<sub>T<\/sub><\/em> := <em>X<sub>1<\/sub><\/em> + \u00b7 \u00b7 \u00b7 + <em>X<sub>T<\/sub><\/em> has expectation <em>E<sub>T<\/sub> \u00b7 EX<sub>1<\/sub><\/em> provided <em>T<\/em> is a stopping time. We prove here that for any 1 < \u03b1 \u2264 2, if T is an arbitrary nonnegative random variable, then <em>S<sub>T<\/sub><\/em> has finite expectation provided that <em>X<sub>1<\/sub><\/em> has finite \u03b1-moment and <em>T<\/em> has finite 1\/(\u03b1 \u2212 1)-moment. We also prove a variant in which <em>T<\/em> is assumed to have a finite exponential moment. These moment conditions are sharp in the sense that for any i.i.d. sequence <em>X<sub>i<\/sub><\/em> violating them, there is a <em>T<\/em> satisfying the given condition for which <em>S<sub>T<\/sub><\/em> (and, in fact, <em>X<sub>T<\/sub><\/em> ) has infinite expectation. An interpretation of this is given in terms of a prophet being more rewarded than a gambler when a certain <em>impatience<\/em> restriction is imposed.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let X1, X2, . . . be independent identically distributed nonnegative random variables. Wald\u2019s identity states that the random sum ST := X1 + \u00b7 \u00b7 \u00b7 + XT has expectation ET \u00b7 EX1 provided T is a stopping time. We prove here that for any 1 < \u03b1 \u2264 2, if T is an [&hellip;]\n<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Institute of Mathematical Statistics","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"Electronic Communications in 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