The tempered discrete Linnik distribution

TL;DR

This paper introduces a tempered discrete Linnik distribution that extends the Poisson-Tweedie law, providing new models connected to stable laws with explicit formulas and special cases.

Contribution

It presents a novel tempered discrete Linnik distribution, generalizing the Poisson-Tweedie law and offering explicit probability functions and identities in law.

Findings

Provides a series of identities in law for the distribution

Offers a manageable expression for the probability function

Analyzes several special cases of the distribution

Abstract

A tempered version of the discrete Linnik distribution is introduced in order to obtain integer-valued distribution families connected to stable laws. The proposal constitutes a generalization of the well-known Poisson-Tweedie law, which is actually a tempered discrete stable law. The features of the new tempered discrete Linnik distribution are explored by providing a series of identities in law - which describe its genesis in terms of mixture and compound Poisson law, as well as in terms of mixture discrete stable law. A manageable expression of the corresponding probability function is also provided and several special cases are analysed.