Distributions based on Stable Mixtures and Gamma-Stable Convolutions

TL;DR

This paper introduces a new family of distributions derived from stable mixtures and gamma convolutions, providing explicit densities and applications to Mittag-Leffler distributions and occupation times in Markov processes.

Contribution

It develops a novel family of distributions based on stable and gamma convolutions, including explicit density formulas and applications to stochastic processes.

Findings

Derived explicit densities for new distribution family

Connected gamma-Linnik convolution to Mittag-Leffler distributions

Linked distributions to occupation times in Markov processes

Abstract

This paper explores mixture distributions induced by a product of the positive stable random variable and a power of another positive random variable. The paper also considers the convolution of the stable density with a gamma density. These two constructs, mixing and convolution, suffice to generate a rich family of distributions. An example is the positive Linnik distribution, which is known to arise from a product involving stable and gamma random variables. We show that gamma-Linnik convolution gives the Mittag-Leffler distribution and the Mittag-Leffler Markov chain associated with the growth of random trees. Building on that, we construct a new family of distributions with explicit densities. A particular choice of parameters for this family yields the Lamperti-type laws associated with occupation times for Markov processes.