On Aggregation of Subcritical Galton–Watson Branching Processes with Regularly Varying Immigration

Abstract

We study an iterated temporal and contemporaneous aggregation of N independent copies of a strongly stationary subcritical Galton-Watson branching process with regularly varying immigration having index α ∈ (0, 2). We show that limits of finite-dimensional distributions of appropriately centered and scaled aggregated partial-sum processes exist when first taking the limit as N → ∞ and then the time scale n → ∞. The limit process is an α-stable process if α ∈ (0, 1) ∪ (1, 2) and a deterministic line with slope 1 if α = 1.

Convergence of partial sum processes to stable processes with application for aggregation of branching processes

“…For each d ∈ Z + , by part (ii) of Proposition D1 in Barczy et al [6], we obtain where (Θ ℓ ) ℓ∈Z + is the (forward) spectral tail process of (X ℓ ) ℓ∈Z + given by Θ ℓ = m ℓ ξ , ℓ ∈ Z + (see, e.g., Theorem E.2), and x + := x ∨ 0, x ∈ R. Indeed, for each d ∈ Z + , Condition (v) of Theorem 2.2 holds trivially, since E(X 1 −E(X 1 )) = 0 in case of α ∈ (1, 4 3 ).…”

“…and hence using that Θ ℓ = m ℓ ξ , ℓ ∈ Z + (see, e.g., (3.7) and (3.8) in Barczy et al [6]), we get exp All in all, the process Z…”

“…for ϑ ∈ R, where (Θ ℓ ) ℓ∈Z + is the (forward) spectral tail process of (X ℓ ) ℓ∈Z + , see Barczy et al [6,Remark 2]. We also remark that, using (14.19) in Sato [34], one can check that (3.1) does not hold in case of α ∈ (1, 4 3 ), which is somewhat unexpected in view of page 171 in Mikosch and Wintenberger [26].…”

“…In the literature the above mentioned procedure is called an iterated and simultaneous aggregation, respectively, see, e.g., Pilipauskaitė and Surgailis [29]. For a review of the literature on aggregation of time series and branching processes, see the Introduction of Barczy et al [6]. Here we mention only our paper Barczy et al [4], where we investigated the limit behaviour of the same aggregation scheme for a stationary multitype Galton-Watson branching process with immigration under third order moment conditions on the offspring and immigration distributions in the iterated and simultaneous cases as well.…”

“…In case of α ∈ (1, 4 3 ) we managed to get rid of the vanishing small value condition assumed in Basrak et al [10,Section 3.2] for the description of the asymptotic behaviour of n k=1 X k as n → ∞. In a companion paper Barczy et al [6] we studied the other iterated, idiosyncratic aggregation scheme, namely, when first taking the limit as the number of copies and then as the time scale tend to infinity. It turned out that the limit distributions are the same for the appropriately scaled and centered aggregated processes as in the present paper.…”