Approximation of Fluctuations in a Sequence of Nearly Critical Branching Processes

Abstract

We consider a sequence of discrete time branching processes with generation-dependent immigration, where the offspring mean tends to its critical value 1. Using a martingale approach, we prove functional limit theorems for suitable normalized fluctuations of the process around its mean when the mean number of immigrating individuals tends to infinity. The limiting processes are deterministically time-changed Wiener processes with three different non-linear time change functions, depending on the behavior of the mean and the variance of the number of immigrants. For the normalized sequence of processes we obtain a deterministic approximation. Consequences related to the maxima and the total progeny of the process will be discussed.

Berry-Esseen bound for nearly critical branching processes with immigration

“…(20) The estimation of the integral J (3) n is not difficult. Collecting (16), (19), (20), and (5), we get the desired result.…”

“…The process {X(t), t ∈ R} turns out to be an Ornstein-Uhlenbeck type process driven by a time changed Wiener process. Further systematic investigations are due to Ispány [7], Györfi et al [5], Li [14], [15] Rahimov [18], [19], Khusanbaev [10], [11], [12], [13] and see also references therein. From these papers, it follows that the necessary norming factor and the possible limit process strongly depend on the various conditions that are supposed to hold for the offspring and immigration processes.…”

On functional limit theorems for branching processes with dependent immigration

“…Motivated by results of Rahimov [21]- [22] on a weak convergence of properly normalized scaled process (1) when immigration is independent, it is a natural to ask about generalization of these results for weakly dependent immigration sequences. In this paper, we focus on weak convergence on Skorokhod topology of the properly normalized scaled process (1) when immigration sequence satisfies some mixing conditions.…”

“…Weakly dependent immigration have also recently been studied by Khusanbaev and Sharipov [13] and established a FLT for a critical branching process under a φ−mixing condition. Rahimov [21]- [22] considered process given by (1) and proved diffusion and fluctuation type limit theorems under assumption that the sequence ε…”

“…The purpose of this paper is to extend some FLT in Rahimov [22] and Sharipov [27] to weakly dependent immigration. In Theorem 2.2, it is proved that scaled random process (1) converges weakly to a deterministic (non-random) function when immigration satisfies ψ−mixing condition.…”