{"paper":{"title":"Free Perpetuities I: Existence, Subordination and Tail Asymptotics","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.PR","authors_text":"Bartosz Ko{\\l}odziejek, Kamil Szpojankowski, Serban Belinschi","submitted_at":"2025-03-13T12:54:49Z","abstract_excerpt":"We study the free analogue of the classical affine fixed-point (or perpetuity) equation\n  \\[\n  \\mathbb{X} \\stackrel{d}{=} \\mathbb{A}^{1/2}\\mathbb{X}\\,\\mathbb{A}^{1/2} + \\mathbb{B},\n  \\] where $\\mathbb{X}$ is assumed to be $*$-free from the pair $(\\mathbb{A},\\mathbb{B})$, with $\\mathbb{A}\\ge 0$ and $\\mathbb{B}=\\mathbb{B}^*$. Our analysis covers both the subcritical regime, where $\\tau(\\mathbb{A})<1$, and the critical case $\\tau(\\mathbb{A})=1$, in which the solution $\\mathbb{X}$ is necessarily unbounded. When $\\tau(\\mathbb{A})=1$, we prove that the series defining $\\mathbb{X}$ converges bilatera"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.10319","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2503.10319/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}