Beyond the Q-process: various ways of conditioningthe multitype Galton-Watson process

Abstract

Conditioning a multitype Galton-Watson process to stay alive into the\nindefinite future leads to what is known as its associated $Q$-process. We show\nthat the same holds true if the process is conditioned to reach a positive\nthreshold or a non-absorbing state. We also demonstrate that the stationary\nmeasure of the $Q$-process, obtained by construction as two successive limits\n(first by delaying the extinction in the original process and next by\nconsidering the long-time behavior of the obtained $Q$-process), is as a matter\nof fact a double limit. Finally, we prove that conditioning a multitype\nbranching process on having an infinite total progeny leads to a process\npresenting the features of a $Q$-process. It does not however coincide with the\noriginal associated $Q$-process, except in the critical regime.\n