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 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.

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.

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.

On a result of Rubin and Vere-Jones concerning subcritical branching processes

1976 ◽  
Vol 13 (04) ◽  
pp. 804-808
Author(s):  
Fred M. Hoppe
Keyword(s):  
Mass Distribution ◽  
Generating Function ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Single Parameter ◽  
Limit Distributions ◽  
Arbitrary Mass ◽  
One To One ◽  
Watson Process ◽  
Conditional Limit

If a subcritical Galton-Watson process is initiated with an arbitrary mass distribution, then it is known that under certain conditions proper conditional limit distributions exist, depending on a single parameter. It is shown here that there is a one-to-one correspondence between these distributions and those arising from the process with a linear offspring probability generating function.

On a result of Rubin and Vere-Jones concerning subcritical branching processes

1976 ◽  
Vol 13 (4) ◽  
pp. 804-808 ◽  
Author(s):  
Fred M. Hoppe
Keyword(s):  
Mass Distribution ◽  
Generating Function ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Single Parameter ◽  
Limit Distributions ◽  
Arbitrary Mass ◽  
One To One ◽  
Watson Process ◽  
Conditional Limit

If a subcritical Galton-Watson process is initiated with an arbitrary mass distribution, then it is known that under certain conditions proper conditional limit distributions exist, depending on a single parameter. It is shown here that there is a one-to-one correspondence between these distributions and those arising from the process with a linear offspring probability generating function.

Functional normalizations for the branching process with infinite mean

1979 ◽  
Vol 16 (03) ◽  
pp. 513-525 ◽  
Author(s):  
Andrew D. Barbour ◽  
H.-J. Schuh
Keyword(s):  
Distribution Function ◽  
Generating Function ◽  
Branching Process ◽  
Analogous Result ◽  
Sufficient Conditions ◽  
Probability Generating Function ◽  
Random Variables ◽  
Watson Process

It is well known that, in a Bienaymé-Galton–Watson process (Zn ) with 1 &lt; m = EZ 1 &lt; ∞ and EZ 1 log Z 1 &lt;∞, the sequence of random variables Znm –n converges a.s. to a non–degenerate limit. When m =∞, an analogous result holds: for any 0&lt; α &lt; 1, it is possible to find functions U such that α n U (Zn ) converges a.s. to a non-degenerate limit. In this paper, some sufficient conditions, expressed in terms of the probability generating function of Z 1 and of its distribution function, are given under which a particular pair (α, U) is appropriate for (Zn ). The most stringent set of conditions reduces, when U (x) x, to the requirements EZ 1 = 1/α, EZ 1 log Z 1 &lt;∞.

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