{"paper":{"title":"On the functional limits for partial sums under stable law","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Kamil Marcin Kosi\\'nski, Khurelbaatar Gonchigdanzan","submitted_at":"2010-06-05T21:52:05Z","abstract_excerpt":"For the partial sums $(S_n)$ of independent random variables we define a stochastic process $s_n(t):=(1/d_n)\\sum_{k \\le [nt]} ({S_k}/{k}-\\mu)$ and prove that $$(1/{\\log N})\\sum_{n\\le N}(1/n)\\mathbf {I}\\left\\{s_n(t)\\le x\\right\\} \\to G_t(x)\\quad \\text{a.s.}$$ if and only if $(1/{\\log N})\\sum_{n\\le N} (1/n)\\mathbb{P}\\left(s_n(t)\\le x\\right) \\to G_t(x)$, for some sequence $(d_n)$ and distribution $G_t$. We also prove an almost sure functional limit theorem for the product of partial sums of i.i.d. positive random variables attracted to an $\\alpha$-stable law with $\\alpha\\in (1,2]$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.1073","kind":"arxiv","version":2},"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"}