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We study the weighted random sum $S_N = \\sum_{i=1}^N C_i X_i$, and its maximum, $M_N = \\sup_{1 \\leq k < N+1} \\sum_{i=1}^k C_i X_i$. These type of sums appear in the analysis of stochastic recursions, including weighted branching processes and autoregressive processes. In particular, we derive conditions under which $$P(M_N"},"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":"1102.0301","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-01T21:28:20Z","cross_cats_sorted":[],"title_canon_sha256":"9a272e11d7e7bab9047bf3495315b10fe3e2289fdaa9bf7f7241babf6245d741","abstract_canon_sha256":"8fac8c9a9ac63eabc3fe4b6cb94b18d2a9491d2a8c3e968a509bb88297188bdf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:57:50.620746Z","signature_b64":"20I3RgaD1XdLaBix3uzbSyKVtNyVBSqrICe1M5gF9EqWHn5ENv+e+ILApx3wPCgr5l7H5w+tNC6EWcwMg/+HCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"575786a24f80d8c226fbd3d6b3c0d05df4e45d058cbe785a8293aa84da7d0823","last_reissued_at":"2026-05-18T03:57:50.620111Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:57:50.620111Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotics for Weighted Random Sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Mariana Olvera-Cravioto","submitted_at":"2011-02-01T21:28:20Z","abstract_excerpt":"Let $\\{X_i\\}$ be a sequence of independent identically distributed random variables with an intermediate regularly varying (IR) right tail $\\bar{F}$. Let $(N, C_1, ..., C_N)$ be a nonnegative random vector independent of the $\\{X_i\\}$ with $N \\in \\mathbb{N} \\cup \\{\\infty\\}$. We study the weighted random sum $S_N = \\sum_{i=1}^N C_i X_i$, and its maximum, $M_N = \\sup_{1 \\leq k < N+1} \\sum_{i=1}^k C_i X_i$. These type of sums appear in the analysis of stochastic recursions, including weighted branching processes and autoregressive processes. 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