{"paper":{"title":"A multidimensional analogue of the arcsine law for the number of positive terms in a random walk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.PR","authors_text":"Dmitry Zaporozhets, Vladislav Vysotsky, Zakhar Kabluchko","submitted_at":"2016-10-10T11:46:17Z","abstract_excerpt":"Consider a random walk $S_i= \\xi_1+\\ldots+\\xi_i$, $i\\in\\mathbb N$, whose increments $\\xi_1,\\xi_2,\\ldots$ are independent identically distributed random vectors in $\\mathbb R^d$ such that $\\xi_1$ has the same law as $-\\xi_1$ and $\\mathbb P[\\xi_1\\in H] = 0$ for every affine hyperplane $H\\subset \\mathbb R^d$. Our main result is the distribution-free formula $$ \\mathbb E \\left[\\sum_{1\\leq i_1 < \\ldots < i_k\\leq n} 1_{\\{0\\notin \\text{conv}(S_{i_1},\\ldots, S_{i_k})\\}}\\right] = 2 \\binom n k \\frac {B(k, d-1) + B(k, d-3) +\\ldots} {2^k k!}, $$ where the $B(k,j)$'s are defined by their generating functio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.02861","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"}