A rate of convergence result for the super-critical Galton-Watson process

1970 ◽  
Vol 7 (2) ◽  
pp. 451-454 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Rate Of Convergence ◽  
Generating Function ◽  
Probability Generating Function ◽  
Random Variable ◽  
Convergence Result ◽  
Convergence Problem ◽  
Offspring Distribution ◽  
Watson Process ◽  
Number Of Authors

Let Z0 = 1, Z1, Z2, ··· denote a super-critical Galton-Watson process whose non-degenerate offspring distribution has probability generating function where 1 < m = EZ1 < ∞. The Galton-Watson process evolves in such a way that the generating function Fn(s) of Znis the nth functional iterate of F(s). The convergence problem for Zn, when appropriately normed, has been studied by quite a number of authors; for an ultimate form see Heyde [2]. However, no information has previously been obtained on the rate of such convergence. We shall here suppose that in which case Wn = m –nZn converges almost surely to a non-degenerate random variable W as n → ∞ (Harris [1], p. 13). It is our object to establish the following result on the rate of convergence of Wn to W.

A rate of convergence result for the super-critical Galton-Watson process

1970 ◽  
Vol 7 (02) ◽  
pp. 451-454 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Rate Of Convergence ◽  
Generating Function ◽  
Probability Generating Function ◽  
Random Variable ◽  
Convergence Result ◽  
Convergence Problem ◽  
Offspring Distribution ◽  
Watson Process ◽  
Number Of Authors

Let Z0 = 1, Z1, Z2, ··· denote a super-critical Galton-Watson process whose non-degenerate offspring distribution has probability generating function where 1 &lt; m = EZ 1 &lt; ∞. The Galton-Watson process evolves in such a way that the generating function Fn (s) of Zn is the nth functional iterate of F(s). The convergence problem for Z n, when appropriately normed, has been studied by quite a number of authors; for an ultimate form see Heyde [2]. However, no information has previously been obtained on the rate of such convergence. We shall here suppose that in which case Wn = m –nZn converges almost surely to a non-degenerate random variable W as n → ∞ (Harris [1], p. 13). It is our object to establish the following result on the rate of convergence of Wn to W.

Functional equations and the Galton-Watson process

1969 ◽  
Vol 1 (1) ◽  
pp. 1-42 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Probability Generating Function ◽  
Single Individual ◽  
Fundamental Assumption ◽  
Offspring Distribution ◽  
Watson Process

In the present exposition we are concerned only with the simple Galton-Watson process, initiated by a single ancestor (Harris (1963), Chapter I). Let denote the probability generating function of the offspring distribution of a single individual. Our fundamental assumption, which holds throughout the sequel, is that fj ≠ 1, j = 0,1,2, …; in particular circumstances it shall be necessary to strengthen this to 0 < f0 ≡ F(0) < 1, which is the relevant assumption when extinction behaviour is to be considered. (Even so, our assumptions will always differ slightly from those of Harris (1963), p. 5.)

Functional equations and the Galton-Watson process

1969 ◽  
Vol 1 (01) ◽  
pp. 1-42 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Probability Generating Function ◽  
Single Individual ◽  
Fundamental Assumption ◽  
Offspring Distribution ◽  
Watson Process

In the present exposition we are concerned only with the simple Galton-Watson process, initiated by a single ancestor (Harris (1963), Chapter I). Letdenote the probability generating function of the offspring distribution of a single individual. Our fundamental assumption, which holds throughout the sequel, is thatfj≠ 1,j= 0,1,2, …; in particular circumstances it shall be necessary to strengthen this to 0 &lt;f0≡F(0) &lt; 1, which is the relevant assumption when extinction behaviour is to be considered. (Even so, our assumptions will always differ slightly from those of Harris (1963), p. 5.)

A note on a functional equation arising in Galton-Watson branching processes

1971 ◽  
Vol 8 (3) ◽  
pp. 589-598 ◽  
Author(s):  
Krishna B. Athreya
Keyword(s):  
Functional Equation ◽  
Continuous Function ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Age Dependent ◽  
Watson Process ◽  
Natural Way

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 < m = h'(1 –) < ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) < 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

A note on a functional equation arising in Galton-Watson branching processes

1971 ◽  
Vol 8 (03) ◽  
pp. 589-598 ◽  
Author(s):  
Krishna B. Athreya
Keyword(s):  
Functional Equation ◽  
Continuous Function ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Age Dependent ◽  
Watson Process ◽  
Natural Way

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 &lt; m = h'(1 –) &lt; ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) &lt; 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

Domains of attraction for the subcritical Galton-Watson branching process

1968 ◽  
Vol 5 (01) ◽  
pp. 216-219 ◽  
Author(s):  
H. Rubin ◽  
D. Vere-Jones
Keyword(s):  
Generating Function ◽  
Branching Process ◽  
Domains Of Attraction ◽  
Offspring Distribution ◽  
Watson Process ◽  
Image Position

Let F(z) = σ fjzj be the generating function for the offspring distribution {fj } from a single ancestor in the usual Galton-Watson process. It is well-known (see Harris [1]) that if Π(z) is the generating function of the distribution of ancestors in the 0th generation, the distribution of offspring at the nth generation has generating function where F n (z), the nth functional iterate of F(z), gives the distribution of offspring at the nth generation from a single ancestor.

On estimating the variance of the offspring distribution in a simple branching process

1974 ◽  
Vol 6 (03) ◽  
pp. 421-433 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Rate Of Convergence ◽  
Central Limit ◽  
Branching Process ◽  
Limit Theory ◽  
Offspring Distribution ◽  
Iterated Logarithm ◽  
Single Realization ◽  
Watson Process ◽  
Limit Result ◽  
Strongly Consistent

A single realization {Z n , 0 ≦n≦N + 1} of a supercritical Galton-Watson process (so called) is considered and it is required to estimate the variance of the offspring distribution. A prospective estimator is proposed, where , and is shown to be strongly consistent on the non-extinction set. A central limit result and an iterated logarithm result are provided to give information on the rate of convergence of the estimator. It is also shown that the estimation results are robust in the sense that they continue to apply unchanged in the case where immigration occurs. Martingale limit theory is employed at each stage in obtaining the limit results.

Inequalities for branching processes

1966 ◽  
Vol 3 (01) ◽  
pp. 261-267 ◽  
Author(s):  
C. R. Heathcote ◽  
E. Seneta
Keyword(s):  
Generating Function ◽  
Conditional Distribution ◽  
Branching Process ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Random Variable ◽  
Expected Time ◽  
Time To Extinction ◽  
Explicit Formulae

Summary If F(s) is the probability generating function of a non-negative random variable, the nth functional iterate Fn (s) = Fn– 1 (F(s)) generates the distribution of the size of the nth generation of a simple branching process. In general it is not possible to obtain explicit formulae for many quantities involving Fn (s), and this paper considers certain bounds and approximations. Bounds are found for the Koenigs-type iterates lim n→∞ m −n {1−Fn (s)}, 0 ≦ s ≦ 1 where m = F′ (1) &lt; 1 and F′′ (1) &lt; ∞; for the expected time to extinction and for the limiting conditional-distribution generating function limn→∞{Fn (s) − Fn (0)} [1 – Fn (0)]–1. Particular attention is paid to the case F(s) = exp {m(s − 1)}.

Domains of attraction for the subcritical Galton-Watson branching process

1968 ◽  
Vol 5 (1) ◽  
pp. 216-219 ◽  
Author(s):  
H. Rubin ◽  
D. Vere-Jones
Keyword(s):  
Generating Function ◽  
Branching Process ◽  
Domains Of Attraction ◽  
Offspring Distribution ◽  
Watson Process ◽  
Image Position

Let F(z) = σ fjzj be the generating function for the offspring distribution {fj} from a single ancestor in the usual Galton-Watson process. It is well-known (see Harris [1]) that if Π(z) is the generating function of the distribution of ancestors in the 0th generation, the distribution of offspring at the nth generation has generating function where Fn(z), the nth functional iterate of F(z), gives the distribution of offspring at the nth generation from a single ancestor.

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