{"ID":2858714,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.06891","arxiv_id":"2510.06891","title":"Multivariate CLT for Lévy processes: convergence rates without moment assumptions","abstract":"We prove that the norm of a $d$-dimensional Lévy process possesses a finite second moment if and only if the convex distance between an appropriately rescaled process at time $t$ and a standard Gaussian vector is integrable in time with respect to the scale-invariant measure $t^{-1} dt$ on $[1,\\infty)$. We further prove that under the standard $\\sqrt{t}$-scaling, the corresponding convex distance is integrable if and only if the norm of the Lévy process has a finite $(2+\\log)$-moment. Both equivalences also hold for the integrability with respect to $t^{-1} dt$ of the multivariate Kolmogorov distance. Our results imply: (I) polynomial Berry-Esseen bounds on the rate of convergence in the convex distance in the CLT for Lévy processes cannot hold without finiteness of $(2+δ)$-moments for some $δ\u003e0$ and (II) integrability of the convex distance with respect to $t^{-1} dt$ in the domain of non-normal attraction cannot occur for any scaling function.","short_abstract":"We prove that the norm of a $d$-dimensional Lévy process possesses a finite second moment if and only if the convex distance between an appropriately rescaled process at time $t$ and a standard Gaussian vector is integrable in time with respect to the scale-invariant measure $t^{-1} dt$ on $[1,\\infty)$. We further prov...","url_abs":"https://arxiv.org/abs/2510.06891","url_pdf":"https://arxiv.org/pdf/2510.06891v1","authors":"[\"Jorge González Cázares\",\"David Kramer-Bang\",\"Aleksandar Mijatović\"]","published":"2025-10-08T11:16:27Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"math.ST\"]","methods":"[]","has_code":false}