Further results on the critical Galton-Watson process with immigration

A. G. Pakes
1972 Journal of the Australian Mathematical Society  
Communicated by P. D. Finch Introduction Consider a Galton-Watson process in which each individual reproduces independently of all others and has probability aj (j = 0,1, • • •) of giving rise to j progeny in the following generation, and in which there is an independent immigration component where bj (j = 0, 1, • • •) is the probability that j individuals enter the population at each generation. Defining X n (n = 0, 1, • • •) to be the population size at the «-th generation, it is known that
more » ... n } defines a Markov chain on the non-negative integers. When |x| < 1, let A(x) = YJ-O^X 1 , B(x) = I f = 0^ and P\ n \x) = £ ; = 0 p\fx J where {p\f} (i,j, n = 0, 1, • • •) are the n-step transition probabilities of the Markov chain {X n }. We shall assume that 0 < a 0 , b 0 < 1. Denote the means of the offspring and immigration distributions by a = A'(I-) and /? = B'(l-) respectively. We always assume that /? < oo and, unless otherwise stated, a = 1. In this case the variance of the offspring distribution is given by 2y = A" (I-) and we assume that 0 < y < oo. Finally, let a = ft/y. Pakes [6] has shown that if YJ=i a jJ 2 Io g/> B "( l ~) < °° t h e n n "P ( oo ~* Mo> (« -» oo) where 0 < ju 0 < oo. For the case where {X n } is irreducible and aperiodic, this result shows it to be null-recurrent when a :g 1 and transient otherwise. In section 2 we shall show that rfP\ n) (x) converges to a function U(x) which is regular in the open unit disc and which generates the invariant measure, {/i,}, of {X n }. Seneta [9] has demonstrated the existence and uniqueness (up to a constant multiple) of an invariant measure under very weak hypotheses. A discussion of the asymptotic behaviour of {p { $} and {//,} is given in section 2. Some results on the asymptotic behaviour of the Green's function G tj = YJ™=OP\"\ which exists under the conditions of theorem 1 below and if a > 1, are given in section 3. It was shown in Pakes [6] that XJn converges weakly to a gamma distributed random variable. A related problem for Y n = ^, = 0 J m , the total number of individuals which have existed in the population up to the n-th generation, is considered in section 4. More specifically, we show that YJn 2 converges weakly 277 use, available at https://www.cambridge.org/core/terms. https://doi.
doi:10.1017/s1446788700013690 fatcat:xocetbpctzfklmmltcqzlytnuu