Almost critical branching processes and limit theorems

Abstract

We study almost critical branching processes with infinitely increasing immigration and prove functional limit theorems for these processes.For each n ∈N, let ξ k j n , ( ) { , k, j ∈ } N and ε k n ( ) { , k ∈ } N be independent collections of independent, nonnegative, integer-valued, identically distributed random variables. We define a sequence of random variables X k n ( ) , k ≥ 0, by the following recurrence relations:Processes thus defined are frequently encountered in the theory of branching processes (see, e.g., [1] and the bibliography therein) and are called branching processes with immigration. If ξ k j n , ( ) is interpreted as the number of descendants of the j th particle in a certain population of particles in the (k -1) th generation, and ε k n ( ) is interpreted as the number of particles that immigrate to the population in the k th generation, then X k n ( ) is the total number of particles in the population in the k th generation.We define a random step process X t n ( ), t ≥ 0 , by setting X t n ( ) = X nt n [ ] ( ) , t ≥ 0 , where a [ ] denotes the integer part of a number a. It is clear that the trajectory of the process X n belongs to the Skorokhod space D 0, ∞ [ ). In what follows, we assume that E ξ 1 1 2 , ( ) n ( ) < ∞ and E ε 1 2 ( ) n ( ) < ∞ for all n ∈N. Denote m n = Eξ 1 1 , ( ) n , σ n 2 = var ξ 1 1 , ( ) n , λ n = Eε 1 ( ) n , b n 2 = var ε 1 ( ) n .In [2], the (almost critical) case where m n = 1 + α/n → 1, α ∈R, and σ n 2 → 0 as n → ∞ and the mean value and variance of the immigration ε k n ( ) converge as n → ∞ to finite values was considered. It was proved that, in this case, the process n -1 X t n ( ) converges weakly in the space D 0, ∞ [ ) to a deterministic process, and a functional limit theorem was established for n -/ 1 2 X t n ( ) ( -EX t n ( )), t ≥ 0 . In [3], sufficient conditions for the convergence of the process X t n ( ) (with corresponding normalization) to a deterministic process were investigated for different conditions on the behavior of m n , λ n , σ n 2 , and b n 2 .In the present paper, we consider the almost critical case where the immigration to a population infinitely increases ( λ n → ∞ as n → ∞) in the mean and investigate the growth rate of the random process X n as n → ∞ and the asymptotic behavior of (properly normalized) X n -EX n as n → ∞. In what follows, T > 0 and γ > 1 are fixed numbers, and I A ( ) denotes the indicator of the event A.

On functional limit theorems for branching processes with dependent immigration

“…In the proof of next theorem we use (10) and divide the proof of Theorem 2.5 into two propositions, which together will imply our result. C3) and (C5) hold, then…”

“…Many authors considered process (1) and related FLT when immigration sequence is i.i.d. (see [15], [31], [30], [16]- [17], [7] - [8], [10]- [11], [2] and references therein). All these results require independence condition of the immigration process.…”