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 , , which are not pure compound Poisson.
Supported by the National Science Centre, Poland, grant no. 2019/35/B/ST1/02421.
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.
"On directional convolution equivalent densities." Electron. J. Probab. 27 1 - 19, 2022. https://doi.org/10.1214/22-EJP790