Which Urbanik class $L_k$, do the hyperbolic and the generalized logistic characteristic functions belong to?

TL;DR

This paper classifies certain selfdecomposable variables derived from Laplace distributions within the Urbanik classes, revealing their specific memberships and implications for ratios of their characteristic functions.

Contribution

It determines the precise Urbanik class memberships of hyperbolic and generalized logistic selfdecomposable variables, expanding understanding of their structural properties.

Findings

Hyperbolic-sine and hyperbolic-cosine variables are in L2 minus L3.

Generalized logistic variables are in L1.

Ratios of these characteristic functions are selfdecomposable.

Abstract

Selfdecomposable variables obtained from series of Laplace (double exponential) variables are objects of this study. We proved that hyperbolic-sine and hyperbolic-cosine variables are in the difference of the Urbanik classes $L_2$ and $L_3$ while generalized logistic variable is at least in the Urbanik class $L_1$. Hence some ratios of those corresponding selfdecomposable characteristic functions are again selfdecomposable.