Mean-square and almost-sure convergence of supercritical age-dependent branching processes

Letting Z(t) be the number of objects alive at time t in a general supercritical age-dependent branching process generated by a single ancestor born at time 0, one achieves (Theorem 1) mean-square convergence of Z(t)/E[Z(t)] provided and , where N(t) is the number of offspring of the initial ancestor born by time t and α is the (positive) Malthusian parameter defined by . If the stronger conditions that (Theorem 2) and hold also, then the convergence is almost-sure. It is of interest that the embedded Galton-Watson process of successive generations need not have a finite mean for the conditions of the above theorems to hold. Similar results are obtained for the age-distribution as well.

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Asymptotic behaviour of immigration-branching processes by general set of types. II: Supercritical branching part

Advances in Applied Probability ◽

10.1017/s000186780004074x ◽

1975 ◽

Vol 7 (03) ◽

pp. 468-494

Author(s):

H. Hering

Keyword(s):

Branching Process ◽

Branching Processes ◽

Almost Sure Convergence ◽

Transition Function ◽

Mean Square ◽

Second Moments ◽

Mean Square Convergence ◽

The Mean ◽

Parameter Dependent ◽

Dependent Probability

We construct an immigration-branching process from an inhomogeneous Poisson process, a parameter-dependent probability distribution of populations and a Markov branching process with homogeneous transition function. The set of types is arbitrary, and the parameter is allowed to be discrete or continuous. Assuming a supercritical branching part with primitive first moments and finite second moments, we prove propositions on the mean square convergence and the almost sure convergence of normalized averaging processes associated with the immigration-branching process.

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The asymptotic frequencies of the types in a multitype age-dependent branching process allowing immigration

Journal of Applied Probability ◽

10.1017/s0021900200118510 ◽

1973 ◽

Vol 10 (03) ◽

pp. 652-658

Author(s):

J. Radcliffe

Keyword(s):

Renewal Process ◽

Branching Process ◽

Almost Sure Convergence ◽

Mean Square ◽

Malthusian Parameter ◽

Age Dependent ◽

The Mean

The mean square and almost sure convergence of W(t) = e–αt Z(t) is proved for a super-critical multitype age-dependent branching process allowing immigration at the epochs of a renewal process. It is shown that the Malthusian parameter, asymptotic frequencies of types and stationary age distributions are the same for the processes with and without immigration.

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The asymptotic frequencies of the types in a multitype age-dependent branching process allowing immigration

Journal of Applied Probability ◽

10.2307/3212784 ◽

1973 ◽

Vol 10 (3) ◽

pp. 652-658 ◽

Cited By ~ 1

Author(s):

J. Radcliffe

Keyword(s):

Renewal Process ◽

Branching Process ◽

Almost Sure Convergence ◽

Mean Square ◽

Malthusian Parameter ◽

Age Dependent ◽

The Mean

The mean square and almost sure convergence of W(t) = e–αtZ(t) is proved for a super-critical multitype age-dependent branching process allowing immigration at the epochs of a renewal process. It is shown that the Malthusian parameter, asymptotic frequencies of types and stationary age distributions are the same for the processes with and without immigration.

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Asymptotic behaviour of immigration-branching processes by general set of types. II: Supercritical branching part

Advances in Applied Probability ◽

10.2307/1426123 ◽

1975 ◽

Vol 7 (3) ◽

pp. 468-494

Author(s):

H. Hering

Keyword(s):

Branching Process ◽

Branching Processes ◽

Almost Sure Convergence ◽

Transition Function ◽

Mean Square ◽

Second Moments ◽

Mean Square Convergence ◽

The Mean ◽

Parameter Dependent ◽

Dependent Probability

We construct an immigration-branching process from an inhomogeneous Poisson process, a parameter-dependent probability distribution of populations and a Markov branching process with homogeneous transition function. The set of types is arbitrary, and the parameter is allowed to be discrete or continuous. Assuming a supercritical branching part with primitive first moments and finite second moments, we prove propositions on the mean square convergence and the almost sure convergence of normalized averaging processes associated with the immigration-branching process.

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Malthusian behaviour of the critical and subcritical age-dependent branching processes with arbitrary state space

Journal of Applied Probability ◽

10.2307/3212465 ◽

1976 ◽

Vol 13 (3) ◽

pp. 455-465

Author(s):

D. I. Saunders

Keyword(s):

State Space ◽

Branching Process ◽

Branching Processes ◽

Rates Of Convergence ◽

Expected Number ◽

Subcritical Case ◽

Arbitrary State ◽

Malthusian Parameter ◽

Age Dependent ◽

Number Of Individuals

For the age-dependent branching process with arbitrary state space let M(x, t, A) be the expected number of individuals alive at time t with states in A given an initial individual at x. Subject to various conditions it is shown that M(x, t, A)e–at converges to a non-trivial limit where α is the Malthusian parameter (α = 0 for the critical case, and is negative in the subcritical case). The method of proof also yields rates of convergence.

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

Journal of Applied Probability ◽

10.2307/3212181 ◽

1971 ◽

Vol 8 (3) ◽

pp. 589-598 ◽

Cited By ~ 18

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.

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

Journal of Applied Probability ◽

10.1017/s002190020003566x ◽

1971 ◽

Vol 8 (03) ◽

pp. 589-598 ◽

Cited By ~ 6

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.

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A limit theorem for a class of supercritical branching processes

Journal of Applied Probability ◽

10.1017/s002190020003610x ◽

1972 ◽

Vol 9 (04) ◽

pp. 707-724 ◽

Cited By ~ 17

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].

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Supercritical age-dependent branching processes with immigration

Journal of Applied Probability ◽

10.2307/3212553 ◽

1974 ◽

Vol 11 (4) ◽

pp. 695-702 ◽

Cited By ~ 9

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.

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