Limiting distributions for the general branching process with immigration

1974 ◽  
Vol 11 (4) ◽  
pp. 809-813 ◽  
Author(s):  
S. D. Durham
Keyword(s):  
Generating Function ◽  
Age Structure ◽  
Branching Process ◽  
Critical Case ◽  
Functional Form ◽  
Asymptotic Distributions ◽  
Moment Conditions ◽  
Cumulant Generating Function ◽  
Age Dependent ◽  
Branching Process With Immigration

For the general age-dependent branching process subject to an arbitrary pattern of immigration asymptotic distributions are studied in the supercritical and critical cases. In the supercritical case a new functional form is given for the limiting cumulant-generating function and in the critical case moment conditions are given under which a randomized gamma limit law prevails. The age-structure is also considered.

Limiting distributions for the general branching process with immigration

1974 ◽  
Vol 11 (04) ◽  
pp. 809-813
Author(s):  
S. D. Durham
Keyword(s):  
Generating Function ◽  
Age Structure ◽  
Branching Process ◽  
Critical Case ◽  
Functional Form ◽  
Asymptotic Distributions ◽  
Moment Conditions ◽  
Cumulant Generating Function ◽  
Age Dependent ◽  
Branching Process With Immigration

For the general age-dependent branching process subject to an arbitrary pattern of immigration asymptotic distributions are studied in the supercritical and critical cases. In the supercritical case a new functional form is given for the limiting cumulant-generating function and in the critical case moment conditions are given under which a randomized gamma limit law prevails. The age-structure is also considered.

Total progeny in a multitype critical age dependent branching process with immigration

1974 ◽  
Vol 11 (3) ◽  
pp. 458-470 ◽  
Author(s):  
Howard J. Weiner
Keyword(s):  
Limit Theorem ◽  
Branching Process ◽  
Cell Types ◽  
Dimensional Case ◽  
One Dimensional ◽  
Age Dependent ◽  
Total Progeny ◽  
Critical Age ◽  
The One ◽  
Branching Process With Immigration

In a multitype critical age dependent branching process with immigration, the numbers of cell types born by t, divided by t2, tends in law to a one-dimensional (degenerate) law whose Laplace transform is explicitily given. The method of proof makes a correspondence between the moments in the m-dimensional case and the one-dimensional case, for which the corresponding limit theorem is known. Other applications are given, a possible relaxation of moment assumptions, and extensions are indicated.

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.

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.

The critical branching process with immigration stopped at zero

1985 ◽  
Vol 22 (01) ◽  
pp. 223-227 ◽  
Author(s):  
B. Gail Ivanoff ◽  
E. Seneta
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Critical Case ◽  
Initial Distribution ◽  
New Form ◽  
Watson Process ◽  
Branching Process With Immigration ◽  
Limit Result ◽  
Critical Branching Process

Limit theorems for the Galton–Watson process with immigration (BPI), where immigration is not permitted when the process is in state 0 (so that this state is absorbing), have been studied for the subcritical and supercritical cases by Seneta and Tavaré (1983). It is pointed out here that, apart from a change of context, the corresponding theorem in the critical case has been obtained by Vatutin (1977). Extensions which follow from a more general form of initial distribution are sketched, including a new form of limit result (7).

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

A multi-type critical age-dependent branching process with immigration

1972 ◽  
Vol 9 (4) ◽  
pp. 697-706 ◽  
Author(s):  
Howard J. Weiner
Keyword(s):  
Gamma Distribution ◽  
Generating Functions ◽  
Branching Process ◽  
Cell Types ◽  
Distribution Law ◽  
One Dimensional ◽  
Age Dependent ◽  
Critical Age ◽  
Branching Process With Immigration

In a multi-type critical age-dependent branching process with immigration, the numbers of cell types alive at time t, each divided by t, as t becomes large, tends to a one-dimensional gamma distribution law. The method of proof employs generating functions and compution of asymptotic moments. Connections with earlier results and extensions are indicated.

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