2014/12/10 by Sophie Pénisson, Pénisson, Sophie · 1 citation
Economics, Econometrics and Finance · Mathematics · #Random Matrices and Applications #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:60F05 #msc:60J80
paper · pdf · doi:10.48550/arxiv.1412.3322
13 pages
arxiv created 2016/03/08 · arxiv updated 2016/03/09
Conditioning a multitype Galton-Watson process to stay alive into the indefinite future leads to what is known as its associated Q-process. We show that the same holds true if the process is conditioned to reach a positive threshold or a non-absorbing state. We also demonstrate that the stationary measure of the Q-process, obtained by construction as two successive limits (first by delaying the extinction in the original process and next by considering the long-time behavior of the obtained Q-process), is as a matter of fact a double limit. Finally, we prove that conditioning a multitype branching process on having an infinite total progeny leads to a process presenting the features of a Q-process. It does not however coincide with the original associated Q-process, except in the critical regime.