{"paper":{"title":"A note on the Kesten--Grincevi\\v{c}ius--Goldie theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Peter Kevei","submitted_at":"2015-12-22T21:09:49Z","abstract_excerpt":"Consider the perpetuity equation $X \\stackrel{\\mathcal{D}}{=} A X + B$, where $(A,B)$ and $X$ on the right-hand side are independent. The Kesten--Grincevi\\v{c}ius--Goldie theorem states that $P \\{ X > x \\} \\sim c x^{-\\kappa}$ if $E A^\\kappa = 1$, $E A^\\kappa \\log_+ A < \\infty$, and $E |B|^\\kappa < \\infty$. We assume that $E |B|^\\nu < \\infty$ for some $\\nu > \\kappa$, and consider two cases (i) $E A^\\kappa = 1$, $E A^\\kappa \\log_+ A = \\infty$; (ii) $E A^\\kappa < 1$, $E A^t = \\infty$ for all $t > \\kappa$. We show that under appropriate additional assumptions on $A$ the asymptotic $P \\{ X > x \\} \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.07262","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"}