{"paper":{"title":"Joint asymptotic distribution of certain path functionals of the reflected process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Aleksandar Mijatovic, Martijn Pistorius","submitted_at":"2013-06-28T08:11:37Z","abstract_excerpt":"Let $\\tau(x)$ be the first time the reflected process $Y$ of a Levy processes $X$ crosses x>0. The main aim of the paper is to investigate the asymptotic dependence of the path functionals: $Y(t) = X(t) - \\inf_{0\\leq s\\leq t}X(s)$, $M(t,x)=\\sup_{0\\leq s\\leq t}Y(s)-x$ and $Z(x)=Y(\\tau(x))-x$. We prove that under Cramer's condition on X(1), the functionals $Y(t)$, $M(t,y)$ and $Z(x+y)$ are asymptotically independent as $\\min\\{t,y,x\\}\\to\\infty$. We also characterise the law of the limiting overshoot $Z(\\infty)$ of the reflected process. If, as $\\min\\{t,x\\}\\to\\infty$, the quantity $t\\te{-\\gamma x}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.6746","kind":"arxiv","version":1},"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"}