Moments for a general branching process in a semi-Markovian environment

1980 ◽  
Vol 17 (2) ◽  
pp. 341-349 ◽  
Author(s):  
Craig Whittaker ◽  
Richard M. Feldman
Keyword(s):  
Population Size ◽  
Discrete Time ◽  
Branching Process ◽  
Markovian Environment ◽  
Iterative Computation ◽  
Second Moments ◽  
Environmental Process

A general branching process is extended to allow life length and reproduction probabilities to depend on randomly changing environmental states. First and second moments of the population size with respect to time are derived assuming that the environmental process is semi-Markovian. The results are similar to Markov-renewal type equations which allow, under discrete time, iterative computation of both moments of the population through time.

Moments for a general branching process in a semi-Markovian environment

1980 ◽  
Vol 17 (02) ◽  
pp. 341-349 ◽  
Author(s):  
Craig Whittaker ◽  
Richard M. Feldman
Keyword(s):  
Population Size ◽  
Discrete Time ◽  
Branching Process ◽  
Markovian Environment ◽  
Iterative Computation ◽  
Second Moments ◽  
Environmental Process

A general branching process is extended to allow life length and reproduction probabilities to depend on randomly changing environmental states. First and second moments of the population size with respect to time are derived assuming that the environmental process is semi-Markovian. The results are similar to Markov-renewal type equations which allow, under discrete time, iterative computation of both moments of the population through time.

Linear growth in near-critical population-size-dependent multitype Galton–Watson processes

1989 ◽  
Vol 26 (3) ◽  
pp. 431-445 ◽  
Author(s):  
Fima C. Klebaner
Keyword(s):  
Population Size ◽  
Discrete Time ◽  
Limit Distribution ◽  
Branching Process ◽  
Linear Growth ◽  
Left Eigenvector ◽  
Size Dependent ◽  
Second Moments ◽  
The Mean ◽  
Critical Process

We consider a multitype population-size-dependent branching process in discrete time. A process is considered to be near-critical if the mean matrices of offspring distributions approach the mean matrix of a critical process as the population size increases. We show that if the second moments of offspring distributions stabilize as the population size increases, and the limiting variances are not too large in comparison with the deviation of the means from criticality, then the extinction probability is less than 1 and the process grows arithmetically fast, in the sense that any linear combination which is not orthogonal to the left eigenvector of the limiting mean matrix grows linearly to a limit distribution. We identify cases when the limiting distribution is gamma. A result on transience of multidimensional Markov chains is also given.

Linear growth in near-critical population-size-dependent multitype Galton–Watson processes

1989 ◽  
Vol 26 (03) ◽  
pp. 431-445
Author(s):  
Fima C. Klebaner
Keyword(s):  
Population Size ◽  
Discrete Time ◽  
Limit Distribution ◽  
Branching Process ◽  
Linear Growth ◽  
Left Eigenvector ◽  
Size Dependent ◽  
Second Moments ◽  
The Mean ◽  
Critical Process

We consider a multitype population-size-dependent branching process in discrete time. A process is considered to be near-critical if the mean matrices of offspring distributions approach the mean matrix of a critical process as the population size increases. We show that if the second moments of offspring distributions stabilize as the population size increases, and the limiting variances are not too large in comparison with the deviation of the means from criticality, then the extinction probability is less than 1 and the process grows arithmetically fast, in the sense that any linear combination which is not orthogonal to the left eigenvector of the limiting mean matrix grows linearly to a limit distribution. We identify cases when the limiting distribution is gamma. A result on transience of multidimensional Markov chains is also given.

Multitype processes with reproduction-dependent immigration

1998 ◽  
Vol 35 (02) ◽  
pp. 281-292
Author(s):  
Ibrahim Rahimov
Keyword(s):  
Limit Theorem ◽  
Discrete Time ◽  
Branching Process ◽  
Martingale Approach ◽  
Offspring Distribution ◽  
Second Moments ◽  
Branching Process With Immigration

The multitype discrete time indecomposable branching process with immigration is considered. Using a martingale approach a limit theorem is proved for such processes when the totality of immigrating individuals at a given time depends on evolution of the processes generating by previously immigrated individuals. Corollaries of the limit theorem are obtained for the cases of finite and infinite second moments of offspring distribution in critical processes.

On the parity of individuals in a branching process

1976 ◽  
Vol 13 (02) ◽  
pp. 219-230 ◽  
Author(s):  
J. Gani ◽  
I. W. Saunders
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Branching Process ◽  
Yeast Cells ◽  
Death Process ◽  
Second Moments ◽  
The Mean ◽  
Watson Process ◽  
Number Of Cells ◽  
Birth Death

This paper is concerned with the parity of a population of yeast cells, each of which may bud, not bud or die. Two multitype models are considered: a Galton-Watson process in discrete time, and its analogous birth-death process in continuous time. The mean number of cells with parity 0, 1, 2, … is obtained in both cases; some simple results are also derived for the second moments of the two processes.

The second moments of the absorption times in the Markov renewal branching process

1978 ◽  
Vol 15 (04) ◽  
pp. 707-714 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Busy Period ◽  
Branching Process ◽  
Queueing Models ◽  
Explicit Formulas ◽  
Absorption Time ◽  
Iterative Computation ◽  
Second Moments ◽  
Number Of Customers

We derive explicit formulas for the second moments of the absorption time matrices in the Markov renewal branching process. These formulas may easily be computationally implemented and are useful in the iterative computation of the semi-Markov matrices, which give the distributions of the duration and of the number of customers served in a busy period of a great variety of complex queueing models.

The second moments of the absorption times in the Markov renewal branching process

1978 ◽  
Vol 15 (4) ◽  
pp. 707-714 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Busy Period ◽  
Branching Process ◽  
Queueing Models ◽  
Explicit Formulas ◽  
Absorption Time ◽  
Iterative Computation ◽  
Second Moments ◽  
Number Of Customers

We derive explicit formulas for the second moments of the absorption time matrices in the Markov renewal branching process. These formulas may easily be computationally implemented and are useful in the iterative computation of the semi-Markov matrices, which give the distributions of the duration and of the number of customers served in a busy period of a great variety of complex queueing models.

A Galton–Watson process with a threshold

2016 ◽  
Vol 53 (2) ◽  
pp. 614-621
Author(s):  
K. B. Athreya ◽  
H.-J. Schuh
Keyword(s):  
Positive Integer ◽  
Population Size ◽  
Discrete Time ◽  
Branching Process ◽  
Branching Processes ◽  
Mean Value ◽  
Size Dependent ◽  
The Mean ◽  
Watson Process ◽  
Critical Branching Process

Abstract In this paper we study a special class of size dependent branching processes. We assume that for some positive integer K as long as the population size does not exceed level K, the process evolves as a discrete-time supercritical branching process, and when the population size exceeds level K, it evolves as a subcritical or critical branching process. It is shown that this process does die out in finite time T. The question of when the mean value E(T) is finite or infinite is also addressed.

Multitype processes with reproduction-dependent immigration

1998 ◽  
Vol 35 (2) ◽  
pp. 281-292
Author(s):  
Ibrahim Rahimov
Keyword(s):  
Limit Theorem ◽  
Discrete Time ◽  
Branching Process ◽  
Martingale Approach ◽  
Offspring Distribution ◽  
Second Moments ◽  
Branching Process With Immigration

The multitype discrete time indecomposable branching process with immigration is considered. Using a martingale approach a limit theorem is proved for such processes when the totality of immigrating individuals at a given time depends on evolution of the processes generating by previously immigrated individuals. Corollaries of the limit theorem are obtained for the cases of finite and infinite second moments of offspring distribution in critical processes.

On the parity of individuals in a branching process

1976 ◽  
Vol 13 (2) ◽  
pp. 219-230 ◽  
Author(s):  
J. Gani ◽  
I. W. Saunders
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Branching Process ◽  
Yeast Cells ◽  
Death Process ◽  
Second Moments ◽  
The Mean ◽  
Watson Process ◽  
Number Of Cells ◽  
Birth Death

This paper is concerned with the parity of a population of yeast cells, each of which may bud, not bud or die. Two multitype models are considered: a Galton-Watson process in discrete time, and its analogous birth-death process in continuous time. The mean number of cells with parity 0, 1, 2, … is obtained in both cases; some simple results are also derived for the second moments of the two processes.

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