Limit theorems for a general critical branching process

1971 ◽  
Vol 8 (1) ◽  
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.

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.

Estimation of the parameters of a branching process from migrating binomial observations

1998 ◽  
Vol 30 (4) ◽  
pp. 948-967 ◽  
Author(s):  
C. Jacob ◽  
J. Peccoud
Keyword(s):  
Confidence Interval ◽  
Asymptotic Behaviour ◽  
Population Size ◽  
Branching Process ◽  
Total Population ◽  
Initial Population ◽  
Asymptotic Confidence Interval ◽  
Total Population Size ◽  
Initial Population Size ◽  
Watson Process

This paper considers a branching process generated by an offspring distribution F with mean m < ∞ and variance σ2 < ∞ and such that, at each generation n, there is an observed δ-migration, according to a binomial law Bpvn*Nnbef which depends on the total population size Nnbef. The δ-migration is defined as an emigration, an immigration or a null migration, depending on the value of δ, which is assumed constant throughout the different generations. The process with δ-migration is a generation-dependent Galton-Watson process, whereas the observed process is not in general a martingale. Under the assumption that the process with δ-migration is supercritical, we generalize for the observed migrating process the results relative to the Galton-Watson supercritical case that concern the asymptotic behaviour of the process and the estimation of m and σ2, as n → ∞. Moreover, an asymptotic confidence interval of the initial population size is given.

scholarly journals Limit theorems for supercritical age-dependent branching processes with neutral immigration

2011 ◽  
Vol 43 (1) ◽  
pp. 276-300 ◽  
Author(s):  
M. Richard
Keyword(s):  
Continuous Spectrum ◽  
Limit Theorems ◽  
Branching Process ◽  
Total Population ◽  
Branching Processes ◽  
Almost Sure Convergence ◽  
Structured Populations ◽  
Total Population Size ◽  
Constant Rate ◽  
Age Dependent

We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate θ, new individuals singly enter the total population and start a new population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagers process: individuals have independent and identically distributed lifetime durations (nonnecessarily exponential) during which they give birth independently at a constant rateb. First, using spine decomposition, we relax previously known assumptions required for almost-sure convergence of the total population size. Then, we consider three models of structured populations: either all immigrants have a different type, or types are drawn in a discrete spectrum or in a continuous spectrum. In each model, the vector (P1,P2,…) of relative abundances of surviving families converges almost surely. In the first model, the limit is the GEM distribution with parameter θ /b.

Asymptotic growth of a class of size-and-age-dependent birth processes

1974 ◽  
Vol 11 (2) ◽  
pp. 248-254 ◽  
Author(s):  
W. A. O'N. Waugh
Keyword(s):  
Population Size ◽  
Branching Process ◽  
Total Population ◽  
Asymptotic Expression ◽  
Branching Processes ◽  
Total Population Size ◽  
Stochastic Population Models ◽  
Age Dependent ◽  
Birth Processes ◽  
Special Case

A class of binary fission stochastic population models is described, in which the fission probabilities may depend on the age of an individual and the total population size. Age-dependent binary branching processes with Erlangian lifelength distributions are a special case. An asymptotic expression for the growth of the population size is developed, which generalizes known theorems about the asymptotic exponential growth of a branching process.

Estimation of the parameters of a branching process from migrating binomial observations

1998 ◽  
Vol 30 (04) ◽  
pp. 948-967 ◽  
Author(s):  
C. Jacob ◽  
J. Peccoud
Keyword(s):  
Confidence Interval ◽  
Asymptotic Behaviour ◽  
Population Size ◽  
Branching Process ◽  
Total Population ◽  
Initial Population ◽  
Asymptotic Confidence Interval ◽  
Total Population Size ◽  
Initial Population Size ◽  
Watson Process

This paper considers a branching process generated by an offspring distribution F with mean m &lt; ∞ and variance σ2 &lt; ∞ and such that, at each generation n, there is an observed δ-migration, according to a binomial law B p v n *N n bef which depends on the total population size N n bef. The δ-migration is defined as an emigration, an immigration or a null migration, depending on the value of δ, which is assumed constant throughout the different generations. The process with δ-migration is a generation-dependent Galton-Watson process, whereas the observed process is not in general a martingale. Under the assumption that the process with δ-migration is supercritical, we generalize for the observed migrating process the results relative to the Galton-Watson supercritical case that concern the asymptotic behaviour of the process and the estimation of m and σ2, as n → ∞. Moreover, an asymptotic confidence interval of the initial population size is given.

scholarly journals Limit theorems for supercritical age-dependent branching processes with neutral immigration

2011 ◽  
Vol 43 (01) ◽  
pp. 276-300 ◽  
Author(s):  
M. Richard
Keyword(s):  
Continuous Spectrum ◽  
Limit Theorems ◽  
Branching Process ◽  
Total Population ◽  
Branching Processes ◽  
Almost Sure Convergence ◽  
Structured Populations ◽  
Total Population Size ◽  
Constant Rate ◽  
Age Dependent

We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate θ, new individuals singly enter the total population and start a new population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagers process: individuals have independent and identically distributed lifetime durations (nonnecessarily exponential) during which they give birth independently at a constant rateb. First, using spine decomposition, we relax previously known assumptions required for almost-sure convergence of the total population size. Then, we consider three models of structured populations: either all immigrants have a different type, or types are drawn in a discrete spectrum or in a continuous spectrum. In each model, the vector (P1,P2,…) of relative abundances of surviving families converges almost surely. In the first model, the limit is the GEM distribution with parameter θ /b.

Asymptotic growth of a class of size-and-age-dependent birth processes

1974 ◽  
Vol 11 (02) ◽  
pp. 248-254 ◽  
Author(s):  
W. A. O'N. Waugh
Keyword(s):  
Population Size ◽  
Branching Process ◽  
Total Population ◽  
Asymptotic Expression ◽  
Branching Processes ◽  
Total Population Size ◽  
Stochastic Population Models ◽  
Age Dependent ◽  
Birth Processes ◽  
Special Case

A class of binary fission stochastic population models is described, in which the fission probabilities may depend on the age of an individual and the total population size. Age-dependent binary branching processes with Erlangian lifelength distributions are a special case. An asymptotic expression for the growth of the population size is developed, which generalizes known theorems about the asymptotic exponential growth of a branching process.

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.

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