On aggregation of multitype Galton–Watson branching processes with 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). Limits of finite dimensional distributions of appropriately centered and scaled aggregated partial sum processes are shown to 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.Recently, Puplinskaitė and Surgailis [21,22] studied iterated aggregation of random coefficient autoregressive processes of order 1 with common innovations and with so-called idiosyncratic innovations, respectively, belonging to the domain of attraction of an α-stable law. Limits 2010 Mathematics Subject Classifications 60J80, 60F05, 60G10, 60G52, 60G70.

Limit Theorems for Branching Processes with Immigration in a Random Environment

“…In deterministic environment for multitype processes, the existence and explicit expression for the moments of order α were subject of Quine [32] for α = 1, 2, and of Barczy et al [2,Lemma 1] for α = 3. Under additional ergodicity conditions, the corresponding result for multitype processes for general α > 0 was proved by Szűcs [36].…”

“…The nonnegative integer d 0 is well defined since P(A 1 = 0) > 0 by the subcriticality assumption (2). Moreover, it represents an accessible atom for the Markov chain (X n ), this makes the chain irreducible, and the stationary distribution unique, see Douc et al [16,Theorem 7.2.1] for instance.…”

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

“…Branching processes, especially Galton-Watson branching processes with immigration, have attracted a lot of attention due to the fact that they are widely used in mathematical biology for modeling the growth of a population in time. In Barczy et al [3], we started to investigate the limit behavior of temporal and contemporaneous aggregations of independent copies of 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 both cases the limit process is a zero-mean Brownian motion with the same covariance function.…”

Statistical inference of subcritical strongly stationary Galton--Watson processes with regularly varying immigration

“…In contrast with those earlier results, we explore the case where the distribution of B is regularly varying with tail index in (1,2), thus having infinite variance. In the sequel, we will assume that µ A ∈ (0, 1), σ 2 A ∈ (0, ∞) and B is regularly varying with tail index α ∈ (1, 2), i.e., lim x→∞ P(B > qx) P(B > x) = q −α for all q ∈ (0, ∞).…”

“…V (1) as n → ∞, where V (1) is an α/2-stable positive random variable, V (2) is a symmetric 2α/3-stable random variable, and V (1) and V (2) are dependent with an explicitly given joint characteristic function, see Theorem 5.1. Concerning the asymptotic behaviour of (a n ) n∈N , note that if x α P(B > x) → 1 as x → ∞, i.e., the distribution of B is asymptotically equivalent to a Pareto distribution with parameter α, then n −1/α a n → (1 − µ A ) −1/α as n → ∞.…”