We show that there is a one-to-one correspondence between invariant measures for the noncritical multitype Galton-Watson process and invariant measures for the single type process with a linear offspring probability generating function. Two corollaries emerge as simple applications, the first being Spitzer's Martin boundary representation, the second giving the asymptotic behaviour of the measures. Both require no extra moment assumptions and are valid for the multitype theory.
"Representations of Invariant Measures on Multitype Galton-Watson Processes." Ann. Probab. 5 (2) 291 - 297, April, 1977. https://doi.org/10.1214/aop/1176995854