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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1405.2674","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-05-12T08:46:17Z","cross_cats_sorted":[],"title_canon_sha256":"958c80740a255b702c70aa025c30df37ce7fbaf2510e5e658a0cec8cd4bfe892","abstract_canon_sha256":"5aac90862bfa011adec9f93ef91e035c3e451e7a01dbe59e2b08f1a2d6156093"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:52:05.001493Z","signature_b64":"UtE0hTyceBeHGCP0HwCqDBlOHwOx4t8+8MuI0Nw1TBnYczI6DGBHzrrvDNlh6m03CpaKkQVKOgJhjDHUOkQiCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3a607e5b8f5d0acd3552b508e07aa543e67401126de66a14f044c6fbf995d288","last_reissued_at":"2026-05-18T02:52:05.001039Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:52:05.001039Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Wald for non-stopping times: The rewards of impatient prophets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexander E. 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