Markov limit of line of decent types in a multitype supercritical branching process

In a multitype (dd types) supercritical positively regular Galton–Watson branching process, let {Xn,Xn−1,…,X0}{Xn,Xn−1,…,X0} denote the types of a randomly chosen (i.e., uniform distribution) individual from the nnth generation and this individual’s nn ancestors. It is shown here that this sequence converges in distribution to a Markov chain {Y0,Y1,…}{Y0,Y1,…} with transition probability matrix (pij)1≤i,j≤d(pij)1≤i,j≤d and having the stationary distribution. We also consider the critical case conditioned on non-extinction.