Asymptotic properties of the stationary measure of a Markov branching process

1973 ◽  
Vol 10 (2) ◽  
pp. 447-450 ◽  
Author(s):  
Y. S. Yang
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Branching Process ◽  
Critical Case ◽  
Asymptotic Properties ◽  
Stationary Measure ◽  
Infinite Variance ◽  
Finite Variance ◽  
Renewal Theorem ◽  
Markov Branching Process

The asymptotic properties of the unique stationary measure of a Markov branching process will be given. In the critical case with finite variance, the result can be deduced from a result for discrete time processes of Kesten, Ney and Spitzer (1966) where the proof makes use of a stronger assumption than the finiteness of variance. For the continuous time case where the stationary measure has an explicit form, we can use the discrete renewal theorem which takes care of the infinite variance case as well.

Asymptotic properties of the stationary measure of a Markov branching process

1973 ◽  
Vol 10 (02) ◽  
pp. 447-450 ◽  
Author(s):  
Y. S. Yang
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Branching Process ◽  
Critical Case ◽  
Asymptotic Properties ◽  
Stationary Measure ◽  
Infinite Variance ◽  
Finite Variance ◽  
Renewal Theorem ◽  
Markov Branching Process

The asymptotic properties of the unique stationary measure of a Markov branching process will be given. In the critical case with finite variance, the result can be deduced from a result for discrete time processes of Kesten, Ney and Spitzer (1966) where the proof makes use of a stronger assumption than the finiteness of variance. For the continuous time case where the stationary measure has an explicit form, we can use the discrete renewal theorem which takes care of the infinite variance case as well.

A branching model by population size dependence

1975 ◽  
Vol 7 (03) ◽  
pp. 495-510
Author(s):  
Carla Lipow
Keyword(s):  
Limit Theorem ◽  
Population Size ◽  
Generating Function ◽  
Continuous Time ◽  
Branching Process ◽  
Size Dependence ◽  
Stationary Measure ◽  
Markov Branching Process ◽  
Branching Model

A continuous-time Markov branching process is modified to allow some dependence of offspring generating function on population size. The model involves a given population size M, below which the offspring generating function is supercritical and above which it is subcritical. Immigration is allowed when the population size is 0. The process has a stationary measure, and an expression for its generating function is found. A limit theorem for the stationary measure as M tends to ∞ is then obtained.

A branching model by population size dependence

1975 ◽  
Vol 7 (3) ◽  
pp. 495-510 ◽  
Author(s):  
Carla Lipow
Keyword(s):  
Limit Theorem ◽  
Population Size ◽  
Generating Function ◽  
Continuous Time ◽  
Branching Process ◽  
Size Dependence ◽  
Stationary Measure ◽  
Markov Branching Process ◽  
Branching Model

A continuous-time Markov branching process is modified to allow some dependence of offspring generating function on population size. The model involves a given population size M, below which the offspring generating function is supercritical and above which it is subcritical. Immigration is allowed when the population size is 0. The process has a stationary measure, and an expression for its generating function is found. A limit theorem for the stationary measure as M tends to ∞ is then obtained.

A continuous time Markov branching model with random environments

1973 ◽  
Vol 5 (1) ◽  
pp. 37-54 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Population Model ◽  
Limit Theorems ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Markov Branching Process ◽  
Branching Model ◽  
Necessary And Sufficient

A population model is constructed which combines the ideas of a discrete time branching process with random environments and a continuous time non-homogeneous Markov branching process. The extinction problem is considered and necessary and sufficient conditions for extinction are determined. Also discussed are limit theorems for what corresponds to the supercritical case.

A continuous time Markov branching model with random environments

1973 ◽  
Vol 5 (01) ◽  
pp. 37-54 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Population Model ◽  
Limit Theorems ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Markov Branching Process ◽  
Branching Model ◽  
Necessary And Sufficient

A population model is constructed which combines the ideas of a discrete time branching process with random environments and a continuous time non-homogeneous Markov branching process. The extinction problem is considered and necessary and sufficient conditions for extinction are determined. Also discussed are limit theorems for what corresponds to the supercritical case.

scholarly journals On Estimation of the Convergence Rate to Invariant Measures in Markov Branching Processes with Possibly Infinite Variance and Allowing Immigration

2021 ◽  
Vol 14 (5) ◽  
pp. 573-583
Author(s):  
Azam A. Imomov ◽  
Keyword(s):  
Generating Functions ◽  
Continuous Time ◽  
Branching Process ◽  
Critical Case ◽  
Invariant Measures ◽  
Branching Processes ◽  
Immigration Law ◽  
Infinite Variance ◽  
Transition Functions ◽  
First Moment

The paper discusses the continuous-time Markov Branching Process allowing Immigration. We are considering a critical case for which the second moment of offspring law and the first moment of immigration law are possibly infinite. Assuming that the nonlinear parts of the appropriate generating functions are regularly varying in the sense of Karamata, we prove theorems on convergence of transition functions of the process to invariant measures. We deduce the speed rate of these convergence providing that slowly varying factors are with remainder

The multitype continuous-time Markov branching process in a periodic environment

1980 ◽  
Vol 12 (1) ◽  
pp. 81-93 ◽  
Author(s):  
B. Klein ◽  
P. D. M. MacDonald
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Cell Kinetics ◽  
Initial Conditions ◽  
Random Variable ◽  
Diurnal Rhythms ◽  
Biological Applications ◽  
Regular Case ◽  
Markov Branching Process ◽  
Periodic Environment

The multitype continuous-time Markov branching process has many biological applications where the environmental factors vary in a periodic manner. Circadian or diurnal rhythms in cell kinetics are an important example. It is shown that in the supercritical positively regular case the proportions of individuals of various types converge in probability to a non-random periodic vector, independent of the initial conditions, while the absolute numbers of individuals of various types converge in probability to that vector multiplied by a random variable whose distribution depends on the initial conditions. It is noted that the proofs are straightforward extensions of the well-known results for a constant environment.

The multitype continuous-time Markov branching process in a periodic environment

1980 ◽  
Vol 12 (01) ◽  
pp. 81-93 ◽  
Author(s):  
B. Klein ◽  
P. D. M. MacDonald
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Cell Kinetics ◽  
Initial Conditions ◽  
Random Variable ◽  
Diurnal Rhythms ◽  
Biological Applications ◽  
Regular Case ◽  
Markov Branching Process ◽  
Periodic Environment

The multitype continuous-time Markov branching process has many biological applications where the environmental factors vary in a periodic manner. Circadian or diurnal rhythms in cell kinetics are an important example. It is shown that in the supercritical positively regular case the proportions of individuals of various types converge in probability to a non-random periodic vector, independent of the initial conditions, while the absolute numbers of individuals of various types converge in probability to that vector multiplied by a random variable whose distribution depends on the initial conditions. It is noted that the proofs are straightforward extensions of the well-known results for a constant environment.

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.

scholarly journals Extinction Times in Multitype Markov Branching Processes

2009 ◽  
Vol 46 (01) ◽  
pp. 296-307 ◽  
Author(s):  
Dominik Heinzmann
Keyword(s):  
Limit Theorem ◽  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Markov Branching Process ◽  
Time To Extinction

In this paper, a distributional approximation to the time to extinction in a subcritical continuous-time Markov branching process is derived. A limit theorem for this distribution is established and the error in the approximation is quantified. The accuracy of the approximation is illustrated in an epidemiological example. Since Markov branching processes serve as approximations to nonlinear epidemic processes in the initial and final stages, our results can also be used to describe the time to extinction for such processes.

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