Applied Modeling Techniques and Data Analysis 2

13On Asymptotic Structure of the Critical Galton-Watson Branching Processes with Infinite Variance and Allowing Immigration

We observe Galton–Watson branching processes with possible immigration. The main results of this chapter are as follows. In the absence of immigration, an integral form of the generating function of the invariant measure in its domain of definition is obtained. In the existing literature, only the “local” form of this function in the neighborhood of point 1 was known (see Slack (1968)). For the processes with immigration, we establish two theorems. The first establishes a formula showing the asymptotic form of the generating function of transition probabilities. This generalizes the result of Pakes (1975), in the sense that he found a similar formula, but only at point 1. In Theorem 13.3, we find the rate of convergence to invariant measures for processes with an infinite variance of the individual transformation law and an infinite mean of the individual immigration law.

13.1. Introduction

Let {X_n, n ∈ ℕ₀ } be the Galton-Watson branching process allowing immigration (GWPI), where N_Q = {0}Uℕ and ℕ = {1,2,...}. This is a homogeneous discrete-time Markov chain with state space S ⊂ ℕ₀ and whose transition probabilities are

p_ij = coefficient of S^j in h(s)(f(s))^l, s ∈ [0,1),

where h(s) = ∑_j∈s hjS^j and f (s) = ∑_j∈s pjS^j are probability generating functions (PGFs). The variable Xn is interpreted as the population size in GWPI at the moment n. An evolution of ...