Limit theorems for the simple branching process allowing immigration, I. The case of finite offspring mean

1979 ◽  
Vol 11 (1) ◽  
pp. 31-62 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Transition Probabilities ◽  
Time Process ◽  
Slowly Varying Function ◽  
Continuous Time Process ◽  
Population Sizes ◽  
Rate Of Decay ◽  
Watson Process ◽  
Reversed Time

This paper presents limit theorems for the population sizes of a Bienaymé–Galton–Watson process allowing immigration. For the non-critical cases it is known that the limit distribution is non-defective iff a logarithmic moment of the immigration distribution is finite. The new results of this paper are concerned with the situation where this moment is infinite and give limit theorems for a certain slowly varying function of the population size. A parallel discussion is given for the critical case and also for the continuous-time process.The methods of the paper are used to give some results on the rate of decay of the transition probabilities and on the growth rate of the stationary measure. These in turn are used to obtain some limit theorems for a reversed-time process.

Limit theorems for the simple branching process allowing immigration, I. The case of finite offspring mean

1979 ◽  
Vol 11 (01) ◽  
pp. 31-62 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Transition Probabilities ◽  
Time Process ◽  
Slowly Varying Function ◽  
Continuous Time Process ◽  
Population Sizes ◽  
Rate Of Decay ◽  
Watson Process ◽  
Reversed Time

This paper presents limit theorems for the population sizes of a Bienaymé–Galton–Watson process allowing immigration. For the non-critical cases it is known that the limit distribution is non-defective iff a logarithmic moment of the immigration distribution is finite. The new results of this paper are concerned with the situation where this moment is infinite and give limit theorems for a certain slowly varying function of the population size. A parallel discussion is given for the critical case and also for the continuous-time process. The methods of the paper are used to give some results on the rate of decay of the transition probabilities and on the growth rate of the stationary measure. These in turn are used to obtain some limit theorems for a reversed-time process.

scholarly journals Deviations for Jumping Times of a Branching Process Indexed by a Poisson Process

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Yanhua Zhang ◽  
Zhenlong Gao
Keyword(s):  
Poisson Process ◽  
Continuous Time ◽  
Branching Process ◽  
Time Process ◽  
Rate Of Growth ◽  
Continuous Time Process ◽  
Watson Process

Consider a continuous time process {Yt=ZNt, t≥0}, where {Zn} is a supercritical Galton–Watson process and {Nt} is a Poisson process which is independent of {Zn}. Let τn be the n-th jumping time of {Yt}, we obtain that the typical rate of growth for {τn} is n/λ, where λ is the intensity of {Nt}. Probabilities of deviations n-1τn-λ-1>δ are estimated for three types of positive δ.

scholarly journals Large deviations for Lotka-Nagaev estimator of a randomly indexed branching process

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5803-5808 ◽  
Author(s):  
Zhenlong Gao ◽  
Lina Qiu
Keyword(s):  
Large Deviations ◽  
Renewal Process ◽  
Continuous Time ◽  
Branching Process ◽  
Asymptotic Properties ◽  
Time Process ◽  
Continuous Time Process ◽  
Watson Process

Consider a continuous time process {Yt=ZNt, t ? 0}, where {Zn} is a supercritical Galton-Watson process and {Nt} is a renewal process which is independent of {Zn}. Firstly, we study the asymptotic properties of the harmonic moments E(Y-rt) of order r > 0 as t ? ?. Then, we obtain the large deviations of the Lotka-Negaev estimator of offspring mean.

Functional limit theorems for the decomposable branching process with two types of particles

2016 ◽  
Vol 26 (2) ◽  
Author(s):  
Valeriy I Afanasiev
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Random Numbers ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Watson Process

AbstractA decomposable Galton - Watson process with two types of particles is considered. Particles of the first type produce equal random numbers of particles of both types, particles of the second type produce particles of the second type only. Under the condition that the total number of the first type particles is equal to

Some limit theorems for Markov branching processes

1973 ◽  
Vol 10 (02) ◽  
pp. 299-306 ◽  
Author(s):  
J. R. Leslie
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Original Process ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
Logarithm Law ◽  
Increasing Process ◽  
Watson Process ◽  
Iterated Logarithm Law

Analogues of the central limit theorem and iterated logarithm law have recently been obtained for the Galton-Watson process; similar results are established in this paper for the temporally homogeneous Markov branching process and for the associated increasing process consisting of the number of splits in the original process up to time t.

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).

Supercritical branching processes with density independent catastrophes

1988 ◽  
Vol 104 (2) ◽  
pp. 413-416 ◽  
Author(s):  
D. R. Grey
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Transition Probabilities ◽  
Branching Processes ◽  
Constant Rate ◽  
Watson Process ◽  
Stationary Transition ◽  
Supercritical Branching Processes ◽  
Independent Identically Distributed ◽  
Independent Poisson Process

A Markov branching process in either discrete time (the Galton–Watson process) or continuous time is modified by the introduction of a process of catastrophes which remove some individuals (and, by implication, their descendants) from the population. The catastrophe process is independent of the reproduction mechanism and takes the form of a sequence of independent identically distributed non-negative integer-valued random variables. In the continuous time case, these catastrophes occur at the points of an independent Poisson process with constant rate. If at any time the size of a catastrophe is at least the current population size, then the population becomes extinct. Thus in both discrete and continuous time we still have a Markov chain with stationary transition probabilities and an absorbing state at zero. Some authors use the term ‘emigration’ as an alternative to ‘catastrophe’.

Some limit theorems for Markov branching processes

1973 ◽  
Vol 10 (2) ◽  
pp. 299-306 ◽  
Author(s):  
J. R. Leslie
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Original Process ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
Logarithm Law ◽  
Increasing Process ◽  
Watson Process ◽  
Iterated Logarithm Law

Analogues of the central limit theorem and iterated logarithm law have recently been obtained for the Galton-Watson process; similar results are established in this paper for the temporally homogeneous Markov branching process and for the associated increasing process consisting of the number of splits in the original process up to time t.

The critical branching process with immigration stopped at zero

1985 ◽  
Vol 22 (1) ◽  
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).

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