{"paper":{"title":"Limiting distribution for the maximal standardized increment of a random walk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Yizao Wang, Zakhar Kabluchko","submitted_at":"2012-11-14T13:33:03Z","abstract_excerpt":"Let $X_1,X_2,...$ be independent identically distributed random variables with $\\mathbb E X_k=0$, $\\mathrm{Var} X_k=1$. Suppose that $\\varphi(t):=\\log \\mathbb E e^{t X_k}<\\infty$ for all $t>-\\sigma_0$ and some $\\sigma_0>0$. Let $S_k=X_1+...+X_k$ and $S_0=0$. We are interested in the limiting distribution of the multiscale scan statistic $$ M_n=\\max_{0\\leq i <j\\leq n} \\frac{S_j-S_i}{\\sqrt{j-i}}. $$ We prove that for an appropriate normalizing sequence $a_n$, the random variable $M_n^2-a_n$ converges to the Gumbel extreme-value law $\\exp\\{-e^{-c x}\\}$. The behavior of $M_n$ depends strongly on t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3301","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"}