Times of a branching process with immigration in varying environment attaining a fixed level

2023/08/07 by Wang, Hua-Ming
#60J10 #60J80 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2308.03614

Abstract

Consider a branching process \Z_n\_{n≥ 0} with immigration in varying environment. For a∈\0,1,2,...\, let C=\n≥0:Z_n=a\ be the collection of times at which the population size of the process attains level a. We give a criterion to determine whether the set C is finite or not. For critical Galton-Watson process, we show that |C∩ [1,n]|/log n→ S in distribution, where S is an exponentially distributed random variable with P(S>t)=e^-t, t>0.

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