2024/05/08 by Jiayan Guo, Guo, Jiayan, Wenming Hong +1
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Diffusion and Search Dynamics #FOS: Mathematics #Primary 60J80 #Probability (math.PR) #Secondary 60F10 #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2405.04835
openalex publication_date 2024/05/08 · openalex created_date 2024/05/11 · openalex updated_date 2026/07/28
We focus on the partial sum Sn=X1+⋯+Xn of the critical branching process with immigration \Xn\, when the offspring ξ is regularly varying with index ν+1 and the immigration η is regularly varying with index δ (0≤ ν<δ<1). The precise large deviation probabilities for Sn are specified, that is, for some appropriate sequences \xn\ and \yn\, uniformly for xn≤ x≤ yn, P(Sn>x)∼ nx-δ/(1+ν)L(x), where L(x) is a slowly varying function. Different from that of the subcritical case, here the upper bound yn is needed. Essentially, this is because the tail probability of the stationary distribution is determined by the offspring or the immigration in the subcritical case. But it is determined by both when the process is critical.