Precise Large Deviations For The Total Population Of Heavy-Tailed Subcritical Branching Process With Immigration

2024/05/07 by Jiayan Guo, Guo, Jiayan, Wenming Hong +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2405.04073

openalex publication_date 2024/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article we focus on the partial sum S_n=X₁+⋯+X_n of the subcritical branching process with immigration \X_n\_n∈\mathbbN₊, under the condition that one of the offspring ξ or immigration η is regularly varying. The tail distribution of S_n is heavily dependent on that of ξ and η, and a precise large deviation probability for S_n is specified. (i)When the tail of offspring ξ is lighter than immigration η, uniformly for x≥ x_n, P(S_n-ES_n>x)∼ c₁nP(η>x) with some constant c₁ and sequence \x_n\, where c₁ is only related to the mean of offspring; (ii) When the tail of immigration η is not heavier than offspring ξ, uniformly for x≥ x_n,P(S_n ES_n>x)∼ c₂nP(ξ>x) with some constant c₂ and sequence \x_n\, where c₂ is related to both the mean of offspring and the mean of immigration.

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