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 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.

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.

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.

Bisexual Galton–Watson branching processes with superadditive mating functions

1986 ◽  
Vol 23 (03) ◽  
pp. 585-600 ◽  
Author(s):  
D. J. Daley ◽  
David M. Hull ◽  
James M. Taylor
Keyword(s):  
Growth Rate ◽  
Branching Process ◽  
Critical Case ◽  
Branching Processes ◽  
Upper And Lower Bounds ◽  
Simple Criterion ◽  
Asymptotic Growth ◽  
Extinction Probabilities ◽  
Mating Function ◽  
Asymptotic Growth Rate

For a bisexual Galton–Watson branching process with superadditive mating function there is a simple criterion for determining whether or not the process becomes extinct with probability 1, namely, that the asymptotic growth rate r should not exceed 1. When extinction is not certain (equivalently, r > 1), simple upper and lower bounds are established for the extinction probabilities. An example suggests that in the critical case that r = 1, some condition like superadditivity is essential for ultimate extinction to be certain. Some illustrative numerical comparisons of particular mating functions are made using a Poisson offspring distribution.

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.

Continuous-time branching processes with decreasing state-dependent immigration

1984 ◽  
Vol 16 (4) ◽  
pp. 697-714 ◽  
Author(s):  
K. V. Mitov ◽  
V. A. Vatutin ◽  
N. M. Yanev
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Continuous Time ◽  
Limit Theorems ◽  
Critical Case ◽  
Branching Processes ◽  
State Dependent ◽  
Different Types ◽  
Continuous Time Branching Processes

This paper deals with continuous-time branching processes which allow a temporally-decreasing immigration whenever the population size is 0. In the critical case the asymptotic behaviour of the probability of non-extinction and of the first two moments is investigated and different types of limit theorems are also proved.

Bisexual Galton–Watson branching processes with superadditive mating functions

1986 ◽  
Vol 23 (3) ◽  
pp. 585-600 ◽  
Author(s):  
D. J. Daley ◽  
David M. Hull ◽  
James M. Taylor
Keyword(s):  
Growth Rate ◽  
Branching Process ◽  
Critical Case ◽  
Branching Processes ◽  
Upper And Lower Bounds ◽  
Simple Criterion ◽  
Asymptotic Growth ◽  
Extinction Probabilities ◽  
Mating Function ◽  
Asymptotic Growth Rate

For a bisexual Galton–Watson branching process with superadditive mating function there is a simple criterion for determining whether or not the process becomes extinct with probability 1, namely, that the asymptotic growth rate r should not exceed 1. When extinction is not certain (equivalently, r > 1), simple upper and lower bounds are established for the extinction probabilities. An example suggests that in the critical case that r = 1, some condition like superadditivity is essential for ultimate extinction to be certain. Some illustrative numerical comparisons of particular mating functions are made using a Poisson offspring distribution.

scholarly journals Asymptotic behavior of critical, irreducible multi-type continuous state and continuous time branching processes with immigration

2016 ◽  
Vol 16 (04) ◽  
pp. 1650008 ◽  
Author(s):  
Mátyás Barczy ◽  
Gyula Pap
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Bessel Process ◽  
Step Functions ◽  
Squared Bessel Process ◽  
Continuous State ◽  
Random Step ◽  
Branching Process With Immigration ◽  
Continuous Time Branching Processes

Under natural assumptions, a Feller type diffusion approximation is derived for critical, irreducible multi-type continuous state and continuous time branching processes with immigration. Namely, it is proved that a sequence of appropriately scaled random step functions formed from a critical, irreducible multi-type continuous state and continuous time branching process with immigration converges weakly towards a squared Bessel process supported by a ray determined by the Perron vector of a matrix related to the branching mechanism of the branching process in question.

The calculation of limit probabilities for denumerable Markov processes from infinitesimal properties

1973 ◽  
Vol 10 (01) ◽  
pp. 84-99 ◽  
Author(s):  
Richard L. Tweedie
Keyword(s):  
Markov Process ◽  
Continuous Time ◽  
Transition Matrix ◽  
Invariant Measures ◽  
Equilibrium Distribution ◽  
Transition Functions ◽  
The Matrix ◽  
Limit Probability ◽  
Derivatives Of ◽  
Regular Chains

The problem considered is that of estimating the limit probability distribution (equilibrium distribution) πof a denumerable continuous time Markov process using only the matrix Q of derivatives of transition functions at the origin. We utilise relationships between the limit vector πand invariant measures for the jump-chain of the process (whose transition matrix we write P∗), and apply truncation theorems from Tweedie (1971) to P∗. When Q is regular, we derive algorithms for estimating πfrom truncations of Q; these extend results in Tweedie (1971), Section 4, from q-bounded processes to arbitrary regular processes. Finally, we show that this method can be extended even to non-regular chains of a certain type.

Continuous-time branching processes with decreasing state-dependent immigration

1984 ◽  
Vol 16 (04) ◽  
pp. 697-714 ◽  
Author(s):  
K. V. Mitov ◽  
V. A. Vatutin ◽  
N. M. Yanev
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Continuous Time ◽  
Limit Theorems ◽  
Critical Case ◽  
Branching Processes ◽  
State Dependent ◽  
Different Types ◽  
Continuous Time Branching Processes

This paper deals with continuous-time branching processes which allow a temporally-decreasing immigration whenever the population size is 0. In the critical case the asymptotic behaviour of the probability of non-extinction and of the first two moments is investigated and different types of limit theorems are also proved.

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