We show that there always exists a sequence of normalizing constants for the supercritical multitype Galton-Watson process so that the normalized sequence converges in probability to a limit which is proper and not identically zero. The Laplace-Stieltjes transform of the limit random variable is characterized as the unique solution under certain conditions of a vector Poincare functional equation.
"Supercritical Multitype Branching Processes." Ann. Probab. 4 (3) 393 - 401, June, 1976. https://doi.org/10.1214/aop/1176996088