A multitype branching process with immigration is considered. A necessary and sufficient condition is found for when the process has a limiting stationary distribution. As a corollary to this result we are able to give necessary and sufficient conditions for when the extinction time for an ordinary multitype process has finite expectation.
"The Multitype Galton-Watson Process with Immigration." Ann. Probab. 1 (6) 947 - 953, December, 1973. https://doi.org/10.1214/aop/1176996802