Extended Sibuya distribution in the Subcritical Markov Branching processes

TL;DR

This paper proves that in subcritical Markov branching processes with a specific mixture of logarithmic distributions, the number of particles alive at any time follows a shifted extended Sibuya distribution, with a precise conditional limit distribution.

Contribution

It introduces the extended Sibuya distribution as the distribution of particles in a specific branching process with a logarithmic mixture mechanism.

Findings

Number of particles follows a shifted extended Sibuya distribution.

Conditional limit distribution is the logarithmic series distribution.

Distribution parameters depend on time t.

Abstract

The subcritical Markov branching process X(t) starting with one particle as the initial condition has the ultimate extinction probability q = 1. The branching mechanism in consideration is defined by the mixture of logarithmic distributions on the nonnegative integers. The purpose of the present paper is to prove that in this case the random number of particles X(t) alive at time t>0 follows the shifted extended Sibuya distribution, with parameters depending on the time $t>0$. The conditional limit probability is exactly the logarithmic series distribution supported by the positive integers.