{"paper":{"title":"Heavy traffic and heavy tails for the maximum of a random walk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Denis Denisov, Johannes Kugler","submitted_at":"2014-03-28T10:16:15Z","abstract_excerpt":"Consider a family of random walks $S_n^{(a)}=X_1^{(a)}+\\cdots+X_n^{(a)}$ with negative drift $\\mathbf E X_1^{(a)}=-a<0$ and finite variance $\\mbox{var}(X_1^{(a)})=\\sigma^2<\\infty$.Let $M^{(a)}=\\max_{n\\ge 0} S_n^{(a)}$ be the maximums of the random walks. The exponential asymptotics $\\mathbf P(aM^{(a)}>x)\\sim e^{-2x/\\sigma^2}$, as $a\\to 0$, were found by Kingman and are known as heavy traffic approximation in the queueing theory. For subexponential random variables the large deviation asymptotics for $\\mathbf P(M^{(a)}>x)\\sim \\frac{1}{a}\\overline F^I(x)$ hold for fixed $a$ as $x\\to\\infty$. In t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.7325","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"}