Exact sampling formulas for multi-type Galton-Watson processes

Exact formulas for the mean and variance of the proportion of different types in a fixed generation of a multi-type Galton-Watson process are derived. The formulas are given in terms of iterates of the probability generating function of the offspring distribution. It is also shown that the sequence of types backwards from a randomly sampled particle in a fixed generation is a non-homogeneous Markov chain where the transition probabilities can be given explicitly, again in terms of probability generating functions. Two biological applications are considered: mutations in mitochondrial DNA and the polymerase chain reaction.