On possible rates of growth of age-dependent branching processes with immigration

It is shown that if ϕ is a given function out of a large class satisfying a certain regularity condition, then a supercritical age-dependent branching process {Z(t)} exists with deterministic immigration and given life-length and family-size distributions such that Z(t)/(eat ϕ(t)) converges in probability to a non-zero constant, a being the appropriate Malthusian parameter. As an easy corollary one discovers the asymptotic behaviour of some processes with random immigration.

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

Journal of Applied Probability ◽

10.1017/s002190020011811x ◽

1974 ◽

Vol 11 (04) ◽

pp. 678-686

Author(s):

Edgar Z. Ganuza ◽

S. D. Durham

Keyword(s):

Branching Process ◽

Branching Processes ◽

Age Distribution ◽

Almost Sure Convergence ◽

Mean Square ◽

Malthusian Parameter ◽

Age Dependent ◽

Mean Square Convergence ◽

Watson Process ◽

Successive Generations

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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Supercritical age-dependent branching processes with immigration

Journal of Applied Probability ◽

10.1017/s0021900200118133 ◽

1974 ◽

Vol 11 (04) ◽

pp. 695-702 ◽

Cited By ~ 1

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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Attainable bounds for expectations

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s144678870001884x ◽

1982 ◽

Vol 33 (3) ◽

pp. 411-420 ◽

Cited By ~ 2

Author(s):

E. Seneta ◽

N. C. Weber

Keyword(s):

Lower Bound ◽

Generating Functions ◽

Branching Process ◽

Simple Technique ◽

Branching Processes ◽

Random Variables ◽

Random Variable ◽

Malthusian Parameter ◽

Age Dependent ◽

Coefficient Of Skewness

AbstractA simple technique for obtaining bounds in terms of means and variances for the expectations of certain functions of random variables in a given class is examined. The bounds given are sharp in the sense that they are attainable by at least one random variable in the class. This technique is applied to obtain bounds for moment generating functions, the coefficient of skewness and parameters associated with branching processes. In particular an improved lower bound for the Malthusian parameter in an age-dependent branching process is derived.

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

Journal of Applied Probability ◽

10.2307/3212551 ◽

1974 ◽

Vol 11 (4) ◽

pp. 678-686 ◽

Cited By ~ 2

Author(s):

Edgar Z. Ganuza ◽

S. D. Durham

Keyword(s):

Branching Process ◽

Branching Processes ◽

Age Distribution ◽

Almost Sure Convergence ◽

Mean Square ◽

Malthusian Parameter ◽

Age Dependent ◽

Mean Square Convergence ◽

Watson Process ◽

Successive Generations

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

Journal of Applied Probability ◽

10.1017/s0021900200104000 ◽

1976 ◽

Vol 13 (03) ◽

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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Branching processes and functional differential equations determining steady-size distributions in cell populations

Journal of Applied Probability ◽

10.1017/s0021900200102529 ◽

1995 ◽

Vol 32 (01) ◽

pp. 1-10

Author(s):

Ziad Taib

Keyword(s):

Differential Equation ◽

Branching Process ◽

Branching Processes ◽

Functional Differential Equations ◽

Cell Populations ◽

Size Distributions ◽

Multitype Branching Process ◽

Functional Differential ◽

Intuitive Interpretation ◽

Infinite Sum

The functional differential equation y′(x) = ay(λx) + by(x) arises in many different situations. The purpose of this note is to show how it arises in some multitype branching process cell population models. We also show how its solution can be given an intuitive interpretation as the probability density function of an infinite sum of independent but not identically distributed random variables.

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A note on the subcritical generalized age-dependent branching process

Journal of Applied Probability ◽

10.2307/3212536 ◽

1976 ◽

Vol 13 (4) ◽

pp. 798-803 ◽

Cited By ~ 2

Author(s):

R. A. Doney

Keyword(s):

Branching Process ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Malthusian Parameter ◽

Age Dependent ◽

The Mean ◽

Mean Function ◽

Necessary And Sufficient

For a subcritical Bellman-Harris process for which the Malthusian parameter α exists and the mean function M(t)∼ aeat as t → ∞, a necessary and sufficient condition for e–at (1 –F(s, t)) to have a non-zero limit is known. The corresponding condition is given for the generalized branching process.

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