{"paper":{"title":"An Exact Asymptotic for the Square Variation of Partial Sum Processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Allison Lewko, Mark Lewko","submitted_at":"2011-06-04T02:57:28Z","abstract_excerpt":"We establish an exact asymptotic formula for the square variation of certain partial sum processes. Let $\\{X_{i}\\}$ be a sequence of independent, identically distributed mean zero random variables with finite variance $\\sigma$ and satisfying a moment condition $\\mathbb{E}[|X_{i}|^{2+\\delta} ] < \\infty$ for some $\\delta > 0$. If we let $\\mathcal{P}_{N}$ denote the set of all possible partitions of the interval $[N]$ into subintervals, then we have that $\\max_{\\pi \\in \\mathcal{P}_{N}} \\sum_{I \\in \\pi} | \\sum_{i\\in I} X_{i}|^2 \\sim 2 \\sigma^2N \\ln \\ln(N)$ holds almost surely. This can be viewed a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.0783","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"}