On the critical Galton-Watson process with immigration

Abstract

Consider a Galton-Watson process in which each individual reproduces independently of all others and has probability aj (j = 0, l, …) of giving rise to j progeny in the following generation and in which there is an independent immigration component where bj (j = 0, l, …) is the probability that j individuals enter the population at each generation. Then letting Xn (n = 0, l, …) be the population size of the n-th generation, it is known (Heathcote [4], [51]) that {Xn} defines a Markov chain on the non-negative integers. Unless otherwise stated, we shall consider only those offspring and immigration distributions that make the Markov chain {Xn} irreducible and aperiodic.

On Long-time Behaviors of States of Galton-Watson Branching Processes Allowing Immigration

“…This result without reference to Seneta has been established also by Pakes [6]. More recent researches on asymptotic properties of process contain in papers [12][13][14] in which the Bernoulli type GWPI was considered, i.e.…”

“…In turn we know that (see [1, p.74] 6) and according to the Basic Lemma of the theory of critical processes 1 − F n (0) ∼ 1/Bn; see e.g. [1, p.19].…”

“…Further long-term properties of states and a problem of existence and uniqueness of invariant measures of GWPI were investigated in papers of Seneta [8,10,11], Pakes [4][5][6][7] and by many other authors. Therein some moment conditions for PGF F (s) and G(s) was required to be satisfied.…”

“…In this case, if in addition to assume that 2B := F ′′ (1) < ∞, properties of S depend on value of parameter λ = α/B: if λ > 1 or λ < 1, then S is transient or null-recurrent accordingly. In the case when λ = 1, Pakes [6] and Zubkov [15] studied necessary and sufficient conditions for a null-recurrence property. Limiting distribution law for critical process {X n } was found first by Seneta [9].…”

The simple harmonic urn

“…The results of Pakes [33] on the critical Galton-Watson process with immigration show that the casino should certainly not add two bonus chips to each stake, for then the process becomes transient, and gambler's ruin will no longer apply.…”