{"paper":{"title":"Bivariate Bernstein-gamma functions, potential measures, and asymptotics of exponential functionals of L\\'evy processes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Martin Minchev, Mladen Savov","submitted_at":"2023-08-22T11:36:16Z","abstract_excerpt":"Let $\\xi$ be a L\\'{e}vy process and $I_\\xi(t):=\\int_{0}^te^{-\\xi_s}\\mathrm{d} s$, $t\\geq 0,$ be the exponential functional of L\\'{e}vy processes on deterministic horizon. Given that $\\lim_{t\\to \\infty}\\xi_t=-\\infty$ we evaluate for general functions $F$ an upper bound on the rate of decay of $\\mathbb{E}\\left(F(I_\\xi(t))\\right)$ based on an explicit integral criterion. When $\\mathbb{E}\\left(\\xi_1\\right)\\in\\left(-\\infty,0\\right)$ and $\\mathbb{P}\\left(\\xi_1>t\\right)$ is regularly varying of index $\\alpha>1$ at infinity, we show that the law of $I_\\xi(t)$, suitably normed and rescaled, converges w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2308.11363","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/2308.11363/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"}