Limit theorems for a branching process with disasters

1976 ◽  
Vol 13 (3) ◽  
pp. 466-475 ◽  
Author(s):  
K. B. Athreya ◽  
N. Kaplan
Keyword(s):  
Population Size ◽  
Limit Theorems ◽  
Branching Process ◽  
Stable Distribution ◽  
Age Distribution ◽  
Random Times

A Bellman–Harris process is considered where the population is subjected to disasters which occur at random times. Each particle alive at the time of a disaster survives it with probability p. In the situation when explosion can occur, several limit theorems are proven. In particular, we prove that the age-distribution converges to the same stable distribution as the Bellman-Harris process and that the population size continues to be asymptotically exponential.

Limit theorems for a branching process with disasters

1976 ◽  
Vol 13 (03) ◽  
pp. 466-475
Author(s):  
K. B. Athreya ◽  
N. Kaplan
Keyword(s):  
Population Size ◽  
Limit Theorems ◽  
Branching Process ◽  
Stable Distribution ◽  
Age Distribution ◽  
Random Times

A Bellman–Harris process is considered where the population is subjected to disasters which occur at random times. Each particle alive at the time of a disaster survives it with probability p. In the situation when explosion can occur, several limit theorems are proven. In particular, we prove that the age-distribution converges to the same stable distribution as the Bellman-Harris process and that the population size continues to be asymptotically exponential.

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.

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.

Exponential growth of a branching process usually implies stable age distribution

1979 ◽  
Vol 16 (03) ◽  
pp. 651-656
Author(s):  
Bo Berndtsson ◽  
Peter Jagers
Keyword(s):  
Population Size ◽  
Exponential Function ◽  
Exponential Growth ◽  
Branching Process ◽  
Age Distribution ◽  
Length Distribution ◽  
Random Variables ◽  
Stable Age Distribution

Start a Bellman–Harris branching process from one or several ancestors, whose ages are identically distributed random variables. Assume that the life-length distribution decays more quickly than exponentially and that the distribution of ages at start does not give too much mass to high ages (in a sense to be made precise). Then, if the expected population size is an exponential function of time, the ages must follow the stable age distribution of the process.

Exponential growth of a branching process usually implies stable age distribution

1979 ◽  
Vol 16 (3) ◽  
pp. 651-656 ◽  
Author(s):  
Bo Berndtsson ◽  
Peter Jagers
Keyword(s):  
Population Size ◽  
Exponential Function ◽  
Exponential Growth ◽  
Branching Process ◽  
Age Distribution ◽  
Length Distribution ◽  
Random Variables ◽  
Stable Age Distribution

Start a Bellman–Harris branching process from one or several ancestors, whose ages are identically distributed random variables. Assume that the life-length distribution decays more quickly than exponentially and that the distribution of ages at start does not give too much mass to high ages (in a sense to be made precise). Then, if the expected population size is an exponential function of time, the ages must follow the stable age distribution of the process.

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.

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 (03) ◽  
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.

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.

Sign in / Sign up

Export Citation Format

Share Document