Time to MRCA for stationary CBI-processes

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.