scholarly journals Limit Properties of Transition Functions of Continuous-Time Markov Branching Processes

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Azam A. Imomov
Keyword(s):  
Continuous Time ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Local Limit ◽  
Markov Branching Process ◽  
Transition Functions ◽  
Basic Lemma ◽  
Invariant Properties ◽  
Local Limit Theorems

Consider the Markov Branching Process with continuous time. Our focus is on the limit properties of transition functions of this process. Using differential analogue of the Basic Lemma we prove local limit theorems for all cases and observe invariant properties of considering process.

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.

scholarly journals Extinction Times in Multitype Markov Branching Processes

2009 ◽  
Vol 46 (1) ◽  
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.

scholarly journals Upper Deviations for Split Times of Branching Processes

2012 ◽  
Vol 49 (4) ◽  
pp. 1134-1143
Author(s):  
Hamed Amini ◽  
Marc Lelarge
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Deviation Probability ◽  
Markov Branching Process ◽  
Split Time

Upper deviation results are obtained for the split time of a supercritical continuous-time Markov branching process. More precisely, we establish the existence of logarithmic limits for the likelihood that the split times of the process are greater than an identified value and determine an expression for the limiting quantity. We also give an estimation for the lower deviation probability of the split times, which shows that the scaling is completely different from the upper deviations.

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.

The extinction time of Markov branching processes

1976 ◽  
Vol 13 (2) ◽  
pp. 345-347 ◽  
Author(s):  
Robert C. Wang
Keyword(s):  
Density Function ◽  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Extinction Time ◽  
Markov Branching Process

This paper considers the expression of expectations and density function of the extinction time for a continuous-time Markov branching process.

Upper Deviations for Split Times of Branching Processes

2012 ◽  
Vol 49 (04) ◽  
pp. 1134-1143
Author(s):  
Hamed Amini ◽  
Marc Lelarge
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Deviation Probability ◽  
Markov Branching Process ◽  
Split Time

Upper deviation results are obtained for the split time of a supercritical continuous-time Markov branching process. More precisely, we establish the existence of logarithmic limits for the likelihood that the split times of the process are greater than an identified value and determine an expression for the limiting quantity. We also give an estimation for the lower deviation probability of the split times, which shows that the scaling is completely different from the upper deviations.

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

scholarly journals Convergence Rates and Limit Theorems for the Dual Markov Branching Process

2017 ◽  
Vol 2017 ◽  
pp. 1-13
Author(s):  
Anthony G. Pakes
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Convergence Rates ◽  
Branching Processes ◽  
Transition Function ◽  
Optimal Convergence ◽  
Markov Branching Process ◽  
Optimal Convergence Rates ◽  
Positive Recurrent

This paper studies aspects of the Siegmund dual of the Markov branching process. The principal results are optimal convergence rates of its transition function and limit theorems in the case that it is not positive recurrent. Additional discussion is given about specifications of the Markov branching process and its dual. The dualising Markov branching processes need not be regular or even conservative.

Explosive Markov branching processes: entrance laws and limiting behaviour

1993 ◽  
Vol 25 (4) ◽  
pp. 737-756 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Recent Work ◽  
Markov Processes ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
General Construction ◽  
Discrete State ◽  
Construction Problem ◽  
Markov Branching Process ◽  
Limiting Behaviour

The supercritical Markov branching process is examined in the case where the minimal version of the process has strictly substochastic transition laws. This provides a nice example of the general construction theory for discrete-state Markov processes.Entrance laws corresponding to the minimal process are characterised. Limit properties of the processes constructed from these entrance laws are examined. All such processes which are honest and cannot hit zero are ergodic. Otherwise these processes are λ-positive and limit theorems conditional on not having left the positive states are given.A connection is made with recent work on the general construction problem when a λ-subinvariant measure is given. The case where immigration is allowed is mentioned.

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