{"paper":{"title":"Pointwise estimates for first passage times of perpetuity sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dariusz Buraczewski, Ewa Damek, Jacek Zienkiewicz","submitted_at":"2015-12-10T21:27:45Z","abstract_excerpt":"We consider first passage times $\\tau_u = \\inf\\{n:\\; Y_n>u\\}$ for the perpetuity sequence $$ Y_n = B_1 + A_1 B_2 + \\cdots + (A_1\\ldots A_{n-1})B_n, $$ where $(A_n,B_n)$ are i.i.d. random variables with values in ${\\mathbb R} ^+\\times {\\mathbb R}$. Recently, a number of limit theorems related to $\\tau_u$ were proved including the law of large numbers, the central limit theorem and large deviations theorems. We obtain a precise asymptotics of the sequence ${\\mathbb P}[\\tau_u = \\log u/\\rho ]$, $\\rho >0$, $u\\to \\infty $ which considerably improves the previous results. There, probabilities ${\\math"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.03449","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"}