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.

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.

The progeny of a branching process

1971 ◽  
Vol 8 (2) ◽  
pp. 407-412 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Generating Function ◽  
Time Distribution ◽  
Branching Process ◽  
Life Time ◽  
Simple Relation ◽  
Age Dependent ◽  
Watson Process ◽  
Image Position

1. Let {Z(t), t ≧ 0} be an age-dependent branching process with offspring generating function and life-time distribution function G(t). Denote by N(t) the progeny of the process, that is the total number of objects which have been born in [0, t], counting the ancestor. (See Section 2 for definitions.) Then in the Galton-Watson process (i.e., when G(t) = 0 for t ≦ 1, G(t) = 1 for t > 1) we have the simple relation Nn = Z0 + Z1 + ··· + Zn, so that the asymptotic behaviour of Nn as n → ∞ follows from a knowledge of the asymptotic behaviour of Zn. In particular, if 1 < m = h'(1) < ∞ and Zn(ω)/E(Zn) → Z(ω) > 0 then also Nn(ω)/E(Nn) → Z(ω) > 0; since E(Zn)/E(Nn) → 1 – m–1 this means that

The progeny of a branching process

1971 ◽  
Vol 8 (02) ◽  
pp. 407-412 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Generating Function ◽  
Time Distribution ◽  
Branching Process ◽  
Life Time ◽  
Simple Relation ◽  
Age Dependent ◽  
Watson Process ◽  
Image Position

1. Let {Z(t), t ≧ 0} be an age-dependent branching process with offspring generating function and life-time distribution function G(t). Denote by N(t) the progeny of the process, that is the total number of objects which have been born in [0, t], counting the ancestor. (See Section 2 for definitions.) Then in the Galton-Watson process (i.e., when G(t) = 0 for t ≦ 1, G(t) = 1 for t &gt; 1) we have the simple relation Nn = Z 0 + Z 1 + ··· + Zn , so that the asymptotic behaviour of Nn as n → ∞ follows from a knowledge of the asymptotic behaviour of Zn . In particular, if 1 &lt; m = h'(1) &lt; ∞ and Zn (ω)/E(Zn ) → Z(ω) &gt; 0 then also Nn (ω)/E(Nn ) → Z(ω) &gt; 0; since E(Zn )/E(Nn ) → 1 – m –1 this means that

scholarly journals The Critical Galton-Watson Process Without Further Power Moments

2007 ◽  
Vol 44 (03) ◽  
pp. 753-769 ◽  
Author(s):  
S. V. Nagaev ◽  
V. Wachtel
Keyword(s):  
Asymptotic Behavior ◽  
Limit Theorem ◽  
Generating Function ◽  
Branching Process ◽  
Alternative Proof ◽  
Conditional Limit Theorem ◽  
Power Moments ◽  
Watson Process ◽  
Conditional Limit

In this paper we prove a conditional limit theorem for a critical Galton-Watson branching process {Z n ; n ≥ 0} with offspring generating function s + (1 − s)L((1 − s)−1), where L(x) is slowly varying. In contrast to a well-known theorem of Slack (1968), (1972) we use a functional normalization, which gives an exponential limit. We also give an alternative proof of Sze's (1976) result on the asymptotic behavior of the nonextinction probability.

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.

scholarly journals On the transient behaviour of a Poisson branching process

1967 ◽  
Vol 7 (4) ◽  
pp. 465-480 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Branching Process ◽  
General Information ◽  
Transient Behaviour ◽  
Offspring Distribution ◽  
Biological Interest ◽  
Watson Process ◽  
Mean Time

SummaryIn general, information concerning the distribution of the time to absorption, T, of a simple branching (Galton-Watson) process for which extinction in finite mean time is certain, is difficult to obtain. The process of greatest biological interest is that for which the offspring distribution is Poisson, having p.g.f. F(s) = em(s-1), m < 1.

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.

Bounds for Certain Branching Processes

1969 ◽  
Vol 6 (01) ◽  
pp. 201-204 ◽  
Author(s):  
Edward Pollak
Keyword(s):  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Offspring Distribution

Summary We consider a branching process for which the offspring distribution has the generating function f(t) and mean f '(1) = m &lt; 1. The probability that a line descended from an individual still survives in generation n is asymptotically of the form cmn. A method is derived whereby good bounds for c may be obtained. This method makes use of the first three moments of the distribution of offspring.

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.)

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