On a functional equation for general branching processes

1973 ◽  
Vol 10 (1) ◽  
pp. 198-205 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Functional Equation ◽  
Integral Equation ◽  
Population Size ◽  
General Class ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variable ◽  
Slow Variation ◽  
Age Dependent ◽  
Class Of Functions

If Z(t) denotes the population size in a Bellman-Harris age-dependent branching process such that a non-denenerate random variable W, then it is known that E(W) = 1 and that ϕ (u) = E(e–uW) satisfies a well-known integral equation. In this situation Athreya [1] has recently found a NASC for E(W |log W| y) <∞, for γ > 0. This paper generalizes Athreya's results in two directions. Firstly a more general class of branching processes is considered; secondly conditions are found for E(W 1 + βL(W)) < ∞ for 0 β < 1, where L is one of a class of functions of slow variation.

On a functional equation for general branching processes

1973 ◽  
Vol 10 (01) ◽  
pp. 198-205 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Functional Equation ◽  
Integral Equation ◽  
Population Size ◽  
General Class ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variable ◽  
Slow Variation ◽  
Age Dependent ◽  
Class Of Functions

If Z(t) denotes the population size in a Bellman-Harris age-dependent branching process such that a non-denenerate random variable W, then it is known that E(W) = 1 and that ϕ (u) = E(e–uW ) satisfies a well-known integral equation. In this situation Athreya [1] has recently found a NASC for E(W |log W| y ) &lt;∞, for γ &gt; 0. This paper generalizes Athreya's results in two directions. Firstly a more general class of branching processes is considered; secondly conditions are found for E(W 1 + βL(W)) &lt; ∞ for 0 β &lt; 1, where L is one of a class of functions of slow variation.

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.

A note on the age-dependent branching process with immigration

1975 ◽  
Vol 12 (01) ◽  
pp. 130-134 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
General Class ◽  
Branching Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Direct Generalization ◽  
Age Dependent ◽  
General Class Of Functions ◽  
Branching Process With Immigration ◽  
Necessary And Sufficient ◽  
Class Of Functions

Let {Z(t)}t0be an age-dependent branching process with immigration. For a general class of functions Φ(x), a necessary and sufficient condition is given for whenE{Φ (Z(t))} &lt;∞. This result is a direct generalization of a theorem proven for the branching process without immigration.

A limit theorem for a class of supercritical branching processes

1972 ◽  
Vol 9 (04) ◽  
pp. 707-724 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
General Class ◽  
Branching Process ◽  
Branching Processes ◽  
Generating Functional ◽  
Number Of Children ◽  
Supercritical Case ◽  
Age Dependent ◽  
Watson Process ◽  
Supercritical Branching Processes ◽  
Convergence In Mean

In the Bellman-Harris (B-H) age-dependent branching process, the birth of a child can occur only at the time of its parent's death. A general class of branching process in which births can occur throughout the lifetime of a parent has been introduced by Crump and Mode. This class shares with the B-H process the property that the generation sizes {ξn } form a Galton-Watson process, and so may be classified into subcritical, critical or supercritical according to the value of m = E{ξ 1}. Crump and Mode showed that, as regards extinction probability, asymptotic behaviour, and for the supercritical case, convergence in mean square of Z(t)/E[Z(t)], as t → ∞, where Z(t) is the population size at time t given one ancestor at t = 0, properties of the B-H process can be extended to this general class. In this paper conditions are found for the convergence in distribution of Z(t)/E{Z(t)} in the supercritical case to a non-degenerate limit distribution. In contrast to the B-H process, these conditions are not the same as those for ξn /mn to have a non-degenerate limit. An integral equation is established for the generating function of Z(t), which is more complicated than the corresponding one for the B-H process and involves the conditional probability generating functional of N(x), x 0, ≧ the number of children born to an individual in the age interval [0, x].

Labelling experiments and phase-age distributions for multiphase branching processes

1973 ◽  
Vol 10 (4) ◽  
pp. 739-747 ◽  
Author(s):  
P. J. Brockwell ◽  
W. H. Kuo
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Joint Probability ◽  
Random Variable ◽  
Cell Labelling ◽  
Single Individual ◽  
Age Dependent ◽  
The Individual ◽  
Number Of Individuals ◽  
Joint Probability Density

A supercritical age-dependent branching process is considered in which the lifespan of each individual is composed of four phases whose durations have joint probability density f(x1, x2, x3, x4). Starting with a single individual of age zero at time zero we consider the asymptotic behaviour as t→ ∞ of the random variable Z(4) (a0,…, an, t) defined as the number of individuals in phase 4 at time t for which the elapsed phase durations Y01,…, Y04,…, Yi1,…, Yi4,…, Yn4 of the individual itself and its first n ancestors satisfy the inequalities Yij ≦ aij, i = 0,…, n, j = 1,…, 4. The application of the results to the analysis of cell-labelling experiments is described. Finally we state an analogous result which defines (conditional on eventual non-extinction of the population) the asymptotic joint distribution of the phase and elapsed phase durations of an individual drawn at random from the population and the phase durations of its ancestors.

Supercritical age-dependent branching processes with immigration

1974 ◽  
Vol 11 (4) ◽  
pp. 695-702 ◽  
Author(s):  
K. B. Athreya ◽  
P. R. Parthasarathy ◽  
G. Sankaranarayanan
Keyword(s):  
Branching Process ◽  
Random Number ◽  
Branching Processes ◽  
Life Time ◽  
Random Variable ◽  
One Dimensional ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Branching Process With Immigration ◽  
Mutually Independent

A branching process with immigration of the following type is considered. For every i, a random number Ni of particles join the system at time . These particles evolve according to a one-dimensional age-dependent branching process with offspring p.g.f. and life time distribution G(t). Assume . Then it is shown that Z(t) e–αt converges in distribution to an extended real-valued random variable Y where a is the Malthusian parameter. We do not require the sequences {τi} or {Ni} to be independent or identically distributed or even mutually independent.

On the asymptotic properties of a supercritical bisexual branching process

1986 ◽  
Vol 23 (03) ◽  
pp. 820-826 ◽  
Author(s):  
J. H. Bagley
Keyword(s):  
Population Size ◽  
General Class ◽  
Population Model ◽  
Branching Process ◽  
Asymptotic Properties ◽  
Almost Sure Convergence ◽  
Random Variable ◽  
Convergence Result ◽  
Bisexual Population

An almost sure convergence result for the normed population size of a bisexual population model is proved. Properties of the limit random variable are deduced. The derivation of similar results for a general class of such processes is discussed.

Asymptotic growth of a class of size-and-age-dependent birth processes

1974 ◽  
Vol 11 (2) ◽  
pp. 248-254 ◽  
Author(s):  
W. A. O'N. Waugh
Keyword(s):  
Population Size ◽  
Branching Process ◽  
Total Population ◽  
Asymptotic Expression ◽  
Branching Processes ◽  
Total Population Size ◽  
Stochastic Population Models ◽  
Age Dependent ◽  
Birth Processes ◽  
Special Case

A class of binary fission stochastic population models is described, in which the fission probabilities may depend on the age of an individual and the total population size. Age-dependent binary branching processes with Erlangian lifelength distributions are a special case. An asymptotic expression for the growth of the population size is developed, which generalizes known theorems about the asymptotic exponential growth of a branching process.

Supercritical age-dependent branching processes with immigration

1974 ◽  
Vol 11 (04) ◽  
pp. 695-702 ◽  
Author(s):  
K. B. Athreya ◽  
P. R. Parthasarathy ◽  
G. Sankaranarayanan
Keyword(s):  
Branching Process ◽  
Random Number ◽  
Branching Processes ◽  
Life Time ◽  
Random Variable ◽  
One Dimensional ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Branching Process With Immigration ◽  
Mutually Independent

A branching process with immigration of the following type is considered. For everyi, a random numberNiof particles join the system at time. These particles evolve according to a one-dimensional age-dependent branching process with offspring p.g.f.and life time distributionG(t). Assume. Then it is shown thatZ(t)e–αtconverges in distribution to an extended real-valued random variableYwhereais the Malthusian parameter. We do not require the sequences {τi} or {Ni} to be independent or identically distributed or even mutually independent.

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