{"paper":{"title":"Asymptotics of the order statistics for a process with a regenerative structure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Natalia Soja-Kukie{\\l}a","submitted_at":"2017-07-05T11:39:35Z","abstract_excerpt":"In the paper, a regenerative process $\\{X_n:n\\in\\mathbb{N}\\}$ with finite mean cycle length is considered. For~$M_n^{(q)}$ denoting the $q$-th largest value in $\\{X_k : 1\\leqslant k \\leqslant n\\}$, we prove that \\begin{equation*} \\sup_{x\\in\\mathbb{R}} \\left|P\\left(M^{(q)}_n\\leqslant x\\right) - G(x)^n \\sum_{k=0}^{q-1}\\frac{\\left(-\\log G(x)^n\\right)^k}{k!}\\gamma_{q,k}(x)\\right| \\to 0,\\quad \\text{as} \\quad n\\to\\infty, \\end{equation*} for $G$ and $\\gamma_{q,k}$ expressed in terms of maxima over the cycle. The result is illustrated with examples."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.01331","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"}