The total progeny of a simple branching process with state-dependent immigration

1983 ◽  
Vol 20 (3) ◽  
pp. 472-481 ◽  
Author(s):  
M. V. Kulkarni ◽  
A. G. Pakes
Keyword(s):  
Population Size ◽  
Limit Theorems ◽  
Branching Process ◽  
State Dependent ◽  
Total Progeny

The paper establishes limit theorems for the total progeny of a simple branching process in which immigration is allowed whenever the population size reaches 0.

The total progeny of a simple branching process with state-dependent immigration

1983 ◽  
Vol 20 (03) ◽  
pp. 472-481 ◽  
Author(s):  
M. V. Kulkarni ◽  
A. G. Pakes
Keyword(s):  
Population Size ◽  
Limit Theorems ◽  
Branching Process ◽  
State Dependent ◽  
Total Progeny

The paper establishes limit theorems for the total progeny of a simple branching process in which immigration is allowed whenever the population size reaches 0.

Some properties of a branching process with group immigration and emigration

1986 ◽  
Vol 18 (3) ◽  
pp. 628-645 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Population Size ◽  
Renewal Process ◽  
Limit Theorems ◽  
Branching Process ◽  
Limiting Distribution ◽  
Immigration And Emigration ◽  
Event Times

Batches of immigrants arrive in a region at event times of a renewal process and individuals grow according to a Bellman-Harris branching process. Tribal emigration allows the possibility that all descendants of a group of immigrants collectively leave the region at some instant.A number of results are derived giving conditions for the existence of a limiting distribution for the population size. These conditions can be given either in terms of the immigration distribution or in terms of the distribution of emigration times. Some limit theorems are obtained when the latter conditions are not fulfilled.

Some limit theorems for the total progeny of a branching process

1971 ◽  
Vol 3 (1) ◽  
pp. 176-192 ◽  
Author(s):  
A. G. Pakes
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Total Progeny

We consider a branching process in which each individual reproduces independently of all others and has probability aj(j = 0, 1, · · ·) of giving rise to j progeny in the following generation. It is assumed, without further comment, that 0 < a0, a0 + a1 < 1.

Continuous-time branching processes with decreasing state-dependent immigration

1984 ◽  
Vol 16 (4) ◽  
pp. 697-714 ◽  
Author(s):  
K. V. Mitov ◽  
V. A. Vatutin ◽  
N. M. Yanev
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Continuous Time ◽  
Limit Theorems ◽  
Critical Case ◽  
Branching Processes ◽  
State Dependent ◽  
Different Types ◽  
Continuous Time Branching Processes

This paper deals with continuous-time branching processes which allow a temporally-decreasing immigration whenever the population size is 0. In the critical case the asymptotic behaviour of the probability of non-extinction and of the first two moments is investigated and different types of limit theorems are also proved.

Bellman-Harris branching processes with a special type of state-dependent immigration

1989 ◽  
Vol 21 (02) ◽  
pp. 270-283 ◽  
Author(s):  
K. V. Mitov ◽  
N. M. Yanev
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Limit Theorems ◽  
Branching Processes ◽  
State Dependent ◽  
Probability Of Extinction ◽  
New Particles

We investigate critical Bellman-Harris processes which allow immigration of new particles whenever the population size is 0. Under some special conditions on the immigration component the asymptotic behaviour of the probability of extinction is obtained and limit theorems are also proved.

Continuous-time branching processes with decreasing state-dependent immigration

1984 ◽  
Vol 16 (04) ◽  
pp. 697-714 ◽  
Author(s):  
K. V. Mitov ◽  
V. A. Vatutin ◽  
N. M. Yanev
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Continuous Time ◽  
Limit Theorems ◽  
Critical Case ◽  
Branching Processes ◽  
State Dependent ◽  
Different Types ◽  
Continuous Time Branching Processes

This paper deals with continuous-time branching processes which allow a temporally-decreasing immigration whenever the population size is 0. In the critical case the asymptotic behaviour of the probability of non-extinction and of the first two moments is investigated and different types of limit theorems are also proved.

Limit theorems for a general critical branching process

1971 ◽  
Vol 8 (01) ◽  
pp. 1-16 ◽  
Author(s):  
Stephen D. Durham
Keyword(s):  
Life Span ◽  
Population Size ◽  
Limit Theorems ◽  
Branching Process ◽  
Total Population ◽  
Life Time ◽  
Multiple Births ◽  
Total Population Size ◽  
Initial Object ◽  
Critical Branching Process

A general branching process begins with an initial object born at time 0. The initial object lives a random length of time and, during its life-time, has offspring which reproduce and die as independent probabilistic copies of the parent. Number and times of births to a parent are random and, once an object is born, its behavior is assumed to be independent of all other objects, independent of total population size and independent of absolute time. The life span of a parent and the number and times its offspring arrive may be interdependent. Multiple births are allowed. The process continues as long as there are objects alive.

Conditional limit theorems for general branching processes

1977 ◽  
Vol 14 (3) ◽  
pp. 451-463 ◽  
Author(s):  
P. J. Green
Keyword(s):  
Population Size ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Conditional Limit Theorems ◽  
Special Case ◽  
Conditional Limit

In this paper we generalise the so-called Yaglom conditional limit theorems to the general branching process counted by the values of a random characteristic, as suggested by Jagers (1974). Even when restricted to the special case of the usual population-size process, our results are stronger than those previously available.

scholarly journals Conditional limit theorems for branching processes

1991 ◽  
Vol 4 (4) ◽  
pp. 263-292 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Asymptotic Behavior ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Random Variable ◽  
Conditional Limit Theorems ◽  
Total Progeny ◽  
Number Of Individuals ◽  
Conditional Limit

Let [ξ(m),m=0,1,2,…] be a branching process in which each individual reproduces independently of the others and has probability pj(j=0,1,2,…) of giving rise to j descendants in the following generation. The random variable ξ(m) is the number of individuals in the mth generation. It is assumed that P{ξ(0)=1}=1. Denote by ρ the total progeny, μ, the time of extinction, and τ, the total number of ancestors of all the individuals in the process. This paper deals with the distributions of the random variables ξ(m), μ and τ under the condition that ρ=n and determines the asymptotic behavior of these distributions in the case where n→∞ and m→∞ in such a way that m/n tends to a finite positive limit.

Conditional limit theorems for general branching processes

1977 ◽  
Vol 14 (03) ◽  
pp. 451-463 ◽  
Author(s):  
P. J. Green
Keyword(s):  
Population Size ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Conditional Limit Theorems ◽  
Special Case ◽  
Conditional Limit

In this paper we generalise the so-called Yaglom conditional limit theorems to the general branching process counted by the values of a random characteristic, as suggested by Jagers (1974). Even when restricted to the special case of the usual population-size process, our results are stronger than those previously available.

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