Time to MRCA for stationary CBI-processes

TL;DR

This paper analyzes the distribution of the time to the MRCA, population sizes, and family structures in stationary continuous state branching processes with immigration, revealing bottleneck effects and ancestral relationships.

Contribution

It introduces a novel approach to compute joint distributions of key population variables in stationary CBI-processes, including the MRCA time and family counts.

Findings

Derived the distribution of the MRCA time and population sizes.

Established the distribution of the number of oldest families.

Identified the bottleneck effect in the population model.

Abstract

Motivated by sample path decomposition of the stationary continuous state branching process with immigration, a general population model is considered using the idea of immortal individual. We compute the joint distribution of the random variables: the time to the most recent common ancestor (MRCA), the size of the current population and the size of the population just before MRCA. We obtain the bottleneck effect as well. The distribution of the number of the oldest families is also established.