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2022 On directional convolution equivalent densities
Kamil Kaleta, Daniel Ponikowski
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Electron. J. Probab. 27: 1-19 (2022). DOI: 10.1214/22-EJP790

Abstract

We propose a definition of directional multivariate subexponential and convolution equivalent densities and find a useful characterization of these notions for a class of integrable and almost radial decreasing functions. We apply this result to show that the density of the absolutely continuous part of the compound Poisson measure built on a given density f is directionally convolution equivalent and inherits its asymptotic behaviour from f if and only if f is directionally convolution equivalent. We also extend this characterization to the densities of more general infinitely divisible distributions on d, d1, which are not pure compound Poisson.

Funding Statement

Supported by the National Science Centre, Poland, grant no. 2019/35/B/ST1/02421.

Acknowledgments

We thank Irmina Czarna, Mateusz Kwaśnicki, René Schilling, Paweł Sztonyk and Toshiro Watanabe for discussions, comments and references. We are grateful to the referee and the editors for careful handling of the paper and helpful comments.

Citation

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Kamil Kaleta. Daniel Ponikowski. "On directional convolution equivalent densities." Electron. J. Probab. 27 1 - 19, 2022. https://doi.org/10.1214/22-EJP790

Information

Received: 15 September 2021; Accepted: 29 April 2022; Published: 2022
First available in Project Euclid: 11 May 2022

Digital Object Identifier: 10.1214/22-EJP790

Subjects:
Primary: 26B99 , 60E05 , 60G50 , 60G51 , 62H05

Keywords: almost radial decreasing function , Compound Poisson measure , cone , Exponential decay , infinitely divisible distribution , isotropic unimodal distribution , Lévy process , multivariate density , random sum , spatial asymptotics , subexponential distribution

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