{"paper":{"title":"Increasing subsequences of random walks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Omer Angel, Rich\\'ard Balka, Yuval Peres","submitted_at":"2014-07-10T16:49:25Z","abstract_excerpt":"Given a sequence of $n$ real numbers $\\{S_i\\}_{i\\leq n}$, we consider the longest weakly increasing subsequence, namely $i_1<i_2<\\dots <i_L$ with $S_{i_k} \\leq S_{i_{k+1}}$ and $L$ maximal. When the elements $S_i$ are i.i.d. uniform random variables, Vershik and Kerov, and Logan and Shepp proved that $\\mathbb{E} L=(2+o(1)) \\sqrt{n}$.\n  We consider the case when $\\{S_i\\}_{i\\leq n}$ is a random walk on $\\mathbb{R}$ with increments of mean zero and finite (positive) variance. In this case, it is well known (e.g., using record times) that the length of the longest increasing subsequence satisfies "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.2860","kind":"arxiv","version":3},"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"}