{"paper":{"title":"A Strong Law of Large Numbers with Applications to Self-Similar Stable Processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Aklilu Zeleke, Erkan Nane, Yimin Xiao","submitted_at":"2008-10-06T20:44:23Z","abstract_excerpt":"Let $p \\in (0, \\infty)$ be a constant and let $\\{\\xi_n\\} \\subset L^p(\\Omega, {\\mathcal F}, \\P)$ be a sequence of random variables. For any integers $m, n \\ge 0$, denote $S_{m, n} = \\sum_{k=m}^{m + n} \\xi_k$. It is proved that, if there exist a nondecreasing function $\\varphi: \\R_+\\to \\R_+$ (which satisfies a mild regularity condition) and an appropriately chosen integer $a\\ge 2$ such that $$ \\sum_{n=0}^\\infty \\sup_{k \\ge 0} \\E\\bigg|\\frac{S_{k, a^n}} {\\varphi(a^n)} \\bigg|^p < \\infty,$$ Then $$ \\lim_{n \\to \\infty} \\frac{S_{0, n}} {\\varphi(n)} = 0\\qquad \\hbox{a.s.} $$ This extends Theorem 1 in Le"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0810.1061","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}