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

We classify the reverse process {Xn} of a multitype Galton-Watson process {Zn}. In the positive recurrent cases we give the stationary measure for {Xn} explicitly, and in the critical case, supposing that all the second moments of Z1 are finite, we establish the convergence in law to a gamma distribution. Limit distributions of {Zcn}, 0 < c < 1, conditioned on Zn, are also given in the subcritical, supercritical and critical cases, respectively. These extend the previous one-type work of W. W. Esty.