A uniform limit theorem and exponential limit law for critical multitype age-dependent branching processes

1978 ◽  
Vol 15 (02) ◽  
pp. 235-242
Author(s):  
Martin I. Goldstein
Keyword(s):  
Limit Theorem ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Uniform Limit ◽  
Second Moment ◽  
Age Dependent ◽  
The Mean ◽  
Image Position ◽  
The Right

Let Z(t) ··· (Z 1(t), …, Zk (t)) be an indecomposable critical k-type age-dependent branching process with generating function F(s, t). Denote the right and left eigenvalues of the mean matrix M by u and v respectively and suppose μ is the vector of mean lifetimes, i.e. Mu = u, vM = v. It is shown that, under second moment assumptions, uniformly for s ∈ ([0, 1] k of the form s = 1 – cu, c a constant. Here vμ is the componentwise product of the vectors and Q[u] is a constant. This result is then used to give a new proof of the exponential limit law.

A uniform limit theorem and exponential limit law for critical multitype age-dependent branching processes

1978 ◽  
Vol 15 (2) ◽  
pp. 235-242 ◽  
Author(s):  
Martin I. Goldstein
Keyword(s):  
Limit Theorem ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Uniform Limit ◽  
Second Moment ◽  
Age Dependent ◽  
The Mean ◽  
Image Position ◽  
The Right

Let Z(t) ··· (Z1(t), …, Zk (t)) be an indecomposable critical k-type age-dependent branching process with generating function F(s, t). Denote the right and left eigenvalues of the mean matrix M by u and v respectively and suppose μ is the vector of mean lifetimes, i.e. Mu = u, vM = v.It is shown that, under second moment assumptions, uniformly for s ∈ ([0, 1]k of the form s = 1 – cu, c a constant. Here vμ is the componentwise product of the vectors and Q[u] is a constant.This result is then used to give a new proof of the exponential limit law.

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 Bounds for the Asymptotic Growth Rate of an Age-Dependent Branching Process

1969 ◽  
Vol 10 (1-2) ◽  
pp. 231-235 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Growth Rate ◽  
Population Size ◽  
Branching Process ◽  
Expected Number ◽  
Lattice Distribution ◽  
Asymptotic Growth ◽  
Age Dependent ◽  
The Mean ◽  
Image Position ◽  
Asymptotic Growth Rate

Let M(t) denote the mean population size at time t (conditional on a single ancestor of age zero at time zero) of a branching process in which the distribution of the lifetime T of an individual is given by Pr {T≦t} =G(t), and in which each individual gives rise (at death) to an expected number A of offspring (1λ A λ ∞). expected number A of offspring (1 < A ∞). Then it is well-known (Harris [1], p. 143) that, provided G(O+)-G(O-) 0 and G is not a lattice distribution, M(t) is given asymptotically by where c is the unique positive value of p satisfying the equation .

Maximal branching processes and ‘long-range percolation’

1970 ◽  
Vol 7 (01) ◽  
pp. 89-98
Author(s):  
John Lamperti
Keyword(s):  
Probability Distribution Function ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Transition Matrices ◽  
A Chain ◽  
Image Position ◽  
The Right

In the first part of this paper, we will consider a class of Markov chains on the non-negative integers which resemble the Galton-Watson branching process, but with one major difference. If there are k individuals in the nth “generation”, and are independent random variables representing their respective numbers of offspring, then the (n + 1)th generation will contain max individuals rather than as in the branching case. Equivalently, the transition matrices Pij of the chains we will study are to be of the form where F(.) is the probability distribution function of a non-negative, integervalued random variable. The right-hand side of (1) is thus the probability that the maximum of i independent random variables distributed by F has the value j. Such a chain will be called a “maximal branching process”.

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

Maximal branching processes and ‘long-range percolation’

1970 ◽  
Vol 7 (1) ◽  
pp. 89-98 ◽  
Author(s):  
John Lamperti
Keyword(s):  
Probability Distribution Function ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Transition Matrices ◽  
A Chain ◽  
Image Position ◽  
The Right

In the first part of this paper, we will consider a class of Markov chains on the non-negative integers which resemble the Galton-Watson branching process, but with one major difference. If there are k individuals in the nth “generation”, and are independent random variables representing their respective numbers of offspring, then the (n + 1)th generation will contain max individuals rather than as in the branching case. Equivalently, the transition matrices Pij of the chains we will study are to be of the form where F(.) is the probability distribution function of a non-negative, integervalued random variable. The right-hand side of (1) is thus the probability that the maximum of i independent random variables distributed by F has the value j. Such a chain will be called a “maximal branching process”.

scholarly journals On asymptotic properties of sub-critical branching processes

1968 ◽  
Vol 8 (4) ◽  
pp. 671-682 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Probability Generating Function ◽  
Image Position

Let Zn be the numer of individuals in the nth generation of a discrete branching process, descended from a single a singel ancestor, for which we put It is well known that the probability generating function of Zn is Fn(s), the n-th functional iterate of F(s), and that if m = EZ1 does not exceed unity, then lim (Harris [1], Chapter 1). In particular, extinction is certain.

An upper bound for the mean of Yaglom's limit

1978 ◽  
Vol 15 (01) ◽  
pp. 199-201 ◽  
Author(s):  
Lawrence S. Evans
Keyword(s):  
Generating Function ◽  
Upper Bound ◽  
Branching Process ◽  
Single Type ◽  
Second Moment ◽  
Yaglom Limit ◽  
The Mean

For a single-type Galton—Watson branching process with mean less than one and finite second moment, we establish an upper bound for the mean of the associated Yaglom limit. This bound is attained if and only if the generating function of the process is linear.

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

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