Reverse processes and some limit theorems of multitype Galton-Watson processes

Abstract

We classify the reverse process { X n} of a multitype Galton-Watson process { Z n}. In the positive recurrent cases we give the stationary measure for { X n} explicitly, and in the critical case, supposing that all the second moments of Z 1 are finite, we establish the convergence in law to a gamma distribution. Limit distributions of { Z cn}, 0 < c < 1, conditioned on Z n, are also given in the subcritical, supercritical and critical cases, respectively. These extend the previous one-type work of W. W. Esty.