Multivariate CLT for Lévy processes: convergence rates without moment assumptions
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866915539101679616 |
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| author | Cázares, Jorge González Kramer-Bang, David Mijatović, Aleksandar |
| author_facet | Cázares, Jorge González Kramer-Bang, David Mijatović, Aleksandar |
| contents | 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 $δ>0$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06891 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multivariate CLT for Lévy processes: convergence rates without moment assumptions Cázares, Jorge González Kramer-Bang, David Mijatović, Aleksandar Probability Statistics Theory 60F05, 60G51 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 $δ>0$ 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. |
| title | Multivariate CLT for Lévy processes: convergence rates without moment assumptions |
| topic | Probability Statistics Theory 60F05, 60G51 |
| url | https://arxiv.org/abs/2510.06891 |