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.
