{"paper":{"title":"Weak convergence of self-normalized partial sums processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Mikl\\'os Cs\\\"org\\H{o}, Zhishui Hu","submitted_at":"2012-04-10T08:09:56Z","abstract_excerpt":"Let $\\{X, X_n, n\\geq 1\\}$ be a sequence of independent identically distributed non-degenerate random variables. Put $S_0=0, S_n = \\sum^n_{i=1} X_i$ and $V_n^2=\\sum^n_{i=1} X_i^2, n\\ge 1.$ A weak convergence theorem is established for the self-normalized partial sums processes $\\{S_{[nt]}/V_n, 0\\le t\\le 1\\}$ when $X$ belongs to the domain of attraction of a stable law with index $\\alpha \\in (0,2]$. The respective limiting distributions of the random variables ${\\max_{1\\le i\\le n}|X_i|}/{S_n}$ and ${\\max_{1\\le i\\le n}|X_i|}/{V_n}$ are also obtained under the same condition."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.2074","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"}