scholarly journals Gaussian fluctuations and a law of the iterated logarithm for Nerman’s martingale in the supercritical general branching process

2021 ◽  
Vol 26 (none) ◽  
Author(s):  
Alexander Iksanov ◽  
Konrad Kolesko ◽  
Matthias Meiners
Keyword(s):  
Branching Process ◽  
Gaussian Fluctuations ◽  
Iterated Logarithm

On estimating the variance of the offspring distribution in a simple branching process

1974 ◽  
Vol 6 (03) ◽  
pp. 421-433 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Rate Of Convergence ◽  
Central Limit ◽  
Branching Process ◽  
Limit Theory ◽  
Offspring Distribution ◽  
Iterated Logarithm ◽  
Single Realization ◽  
Watson Process ◽  
Limit Result ◽  
Strongly Consistent

A single realization {Z n , 0 ≦n≦N + 1} of a supercritical Galton-Watson process (so called) is considered and it is required to estimate the variance of the offspring distribution. A prospective estimator is proposed, where , and is shown to be strongly consistent on the non-extinction set. A central limit result and an iterated logarithm result are provided to give information on the rate of convergence of the estimator. It is also shown that the estimation results are robust in the sense that they continue to apply unchanged in the case where immigration occurs. Martingale limit theory is employed at each stage in obtaining the limit results.

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.

scholarly journals Asymptotic behaviour near extinction of continuous-state branching processes

2016 ◽  
Vol 53 (2) ◽  
pp. 381-391
Author(s):  
Gabriel Berzunza ◽  
Juan Carlos Pardo
Keyword(s):  
Asymptotic Behaviour ◽  
Branching Process ◽  
Branching Processes ◽  
Extinction Time ◽  
Iterated Logarithm ◽  
Continuous State ◽  
Continuous State Branching Process ◽  
Near Extinction

AbstractIn this paper we study the asymptotic behaviour near extinction of (sub-)critical continuous-state branching processes. In particular, we establish an analogue of Khintchine's law of the iterated logarithm near extinction time for a continuous-state branching process whose branching mechanism satisfies a given condition.

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.

On estimating the variance of the offspring distribution in a simple branching process

1974 ◽  
Vol 6 (3) ◽  
pp. 421-433 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Rate Of Convergence ◽  
Central Limit ◽  
Branching Process ◽  
Limit Theory ◽  
Offspring Distribution ◽  
Iterated Logarithm ◽  
Single Realization ◽  
Watson Process ◽  
Limit Result ◽  
Strongly Consistent

A single realization {Zn, 0 ≦n≦N + 1} of a supercritical Galton-Watson process (so called) is considered and it is required to estimate the variance of the offspring distribution. A prospective estimator is proposed, where , and is shown to be strongly consistent on the non-extinction set. A central limit result and an iterated logarithm result are provided to give information on the rate of convergence of the estimator. It is also shown that the estimation results are robust in the sense that they continue to apply unchanged in the case where immigration occurs. Martingale limit theory is employed at each stage in obtaining the limit results.

Some Contributions to the Theory of Near-Critical Bisexual Branching Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 492-505
Author(s):  
M. Molina ◽  
M. Mota ◽  
A. Ramos
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Positive Probability ◽  
Limit Laws ◽  
Bisexual Branching Processes ◽  
Probabilistic Evolution

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

scholarly journals Demographic Dynamics in Multitype Populations with Migrations

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 246
Author(s):  
Manuel Molina-Fernández ◽  
Manuel Mota-Medina
Keyword(s):  
Mathematical Modeling ◽  
Population Dynamics ◽  
Branching Process ◽  
Probability Model ◽  
Biological Systems ◽  
Research Work ◽  
General Setting ◽  
Process Theory ◽  
Multitype Branching Process ◽  
Demographic Dynamics

This research work deals with mathematical modeling in complex biological systems in which several types of individuals coexist in various populations. Migratory phenomena among the populations are allowed. We propose a class of mathematical models to describe the demographic dynamics of these type of complex systems. The probability model is defined through a sequence of random matrices in which rows and columns represent the various populations and the several types of individuals, respectively. We prove that this stochastic sequence can be studied under the general setting provided by the multitype branching process theory. Probabilistic properties and limiting results are then established. As application, we present an illustrative example about the population dynamics of biological systems formed by long-lived raptor colonies.

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