Fluctuation limit of branching processes with immigration and estimation of the means

Abstract

We investigate a sequence of Galton-Watson branching processes with immigration, where the offspring mean tends to its critical value 1 and the offspring variance tends to 0. It is shown that the fluctuation limit is an Ornstein-Uhlenbeck-type process. As a consequence, in contrast to the case in which the offspring variance tends to a positive limit, it transpires that the conditional least-squares estimator of the offspring mean is asymptotically normal. The norming factor is n 3/2 , in contrast to both the subcritical case, in which it is n 1/2 , and the nearly critical case with positive limiting offspring variance, in which it is n.

Efficient Estimation of Auto-Regression Parameters and Innovation Distributions for Semiparametric Integer-Valued AR(p) Models

“…For p =1, inference in the non‐stationary INAR( p ) model, i.e. θ 1 ≈1, has been discussed in Ispány et al. (2003a, b, 2005) and Drost et al.…”

Occupation Time Fluctuations of Weakly Degenerate Branching Systems

“…Furthermore, by using the inequality 1 − e −x ≤ x 1/8 for x ≥ 0 and the similar argument to that in [6, p.29 lines [8][9][10][11][12][13][14][15], we arrive at (3.51). The details are omitted.…”

“…Because of the sub-critical branching laws at positive ages, a fixed (d, α, δ, γ)-branching particle system with δ > 0 will go to local extinction as time elapses. To overcome this difficulty, we borrow the idea of nearly critical branching processes (see [15,19,26]) and consider a sequence of (d, α, δ n , γ)-branching particle systems with δ n → 0 as n → ∞. We study the limit process of the re-scaled occupation time fluctuations…”

On central limit theorems for branching processes with dependent immigration

“…More precisely, they studied weak convergence of n −1 W (n) under more general situation that there exists a random vari- Badalbaev and Zubkov [5] for the sequence of special random processes (including branching processes with immigration) proved a limit theorem which contains results of [29] and [1] as a special case. Concerning functional limit theorems for (1), we refer to Wei and Winnicki [46], Sriram [43], Li [25], [24], Ispány et al [14], [15], Khusanbaev [19], [21], Iksanov and Kabluchko [13] and see references therein. We refer to papers [18], [20], [39] where the rates of convergence in central limit theorem for (1) were studied.…”