Some asymptotic results for multiphase branching processes

1975 ◽  
Vol 7 (2) ◽  
pp. 251-251
Author(s):  
P. D. M. Macdonald
Keyword(s):  
Branching Processes ◽  
Asymptotic Results

Parametric inference in Markov branching processes with time-dependent random immigration rate

1985 ◽  
Vol 22 (03) ◽  
pp. 503-517
Author(s):  
Helmut Pruscha
Keyword(s):  
Point Processes ◽  
Traditional Approach ◽  
Branching Processes ◽  
External Source ◽  
Time Dependent ◽  
Parametric Inference ◽  
Asymptotic Results ◽  
Subcritical Case ◽  
Immigration Rate ◽  
Pearson Type

The present paper deals with continuous-time Markov branching processes allowing immigration. The immigration rate is allowed to be random and time-dependent where randomness may stem from an external source or from state-dependence. Unlike the traditional approach, we base the analysis of these processes on the theory of multivariate point processes. Using the tools of this theory, asymptotic results on parametric inference are derived for the subcritical case. In particular, the limit distributions of some parametric estimators and of Pearson-type statistics for testing simple and composite hypotheses are established.

Sur une procédure de branchement déterministe et ses dérivées aléatoires

1994 ◽  
Vol 31 (02) ◽  
pp. 333-347
Author(s):  
Thierry Huillet ◽  
Andrzej Kłopotowski
Keyword(s):  
Branching Processes ◽  
Asymptotic Results ◽  
Renewal Equations ◽  
Age Dependent ◽  
Probabilistic Version

This paper is concerned with the description of both a deterministic and stochastic branching procedure. The renewal equations for the deterministic branching population are first derived which allow for asymptotic results on the ‘number' and ‘generation' processes. A probabilistic version of these processes is then studied which presents some discrepancy with the standard Harris age-dependent branching processes.

Sur une procédure de branchement déterministe et ses dérivées aléatoires

1994 ◽  
Vol 31 (2) ◽  
pp. 333-347 ◽  
Author(s):  
Thierry Huillet ◽  
Andrzej Kłopotowski
Keyword(s):  
Branching Processes ◽  
Asymptotic Results ◽  
Renewal Equations ◽  
Age Dependent ◽  
Probabilistic Version

This paper is concerned with the description of both a deterministic and stochastic branching procedure. The renewal equations for the deterministic branching population are first derived which allow for asymptotic results on the ‘number' and ‘generation' processes. A probabilistic version of these processes is then studied which presents some discrepancy with the standard Harris age-dependent branching processes.

scholarly journals Asymptotics of fluctuations in Crump‒Mode‒Jagers processes: the lattice case

2018 ◽  
Vol 50 (A) ◽  
pp. 141-171
Author(s):  
Svante Janson
Keyword(s):  
Generating Function ◽  
Branching Processes ◽  
Age Distribution ◽  
Symbolic Calculus ◽  
Asymptotic Results ◽  
Absolute Value ◽  
Malthusian Parameter ◽  
Asymptotically Normal ◽  
Complex Roots ◽  
Degenerate Cases

Abstract Consider a supercritical Crump‒Jagers process in which all births are at integer times (the lattice case). Let μ̂(z) be the generating function of the intensity of the offspring process, and consider the complex roots of μ̂(z)=1. The root of smallest absolute value is e-α=1∕m, where α>0 is the Malthusian parameter; let γ* be the root of second smallest absolute value. Subject to some technical conditions, the second-order fluctuations of the age distribution exhibit one of three types of behaviour: (i) when γ*>e-α∕2=m-1∕2, they are asymptotically normal; (ii) when γ*=e-α∕2, they are still asymptotically normal, but with a larger variance; and (iii) when γ*<e-α∕2, the fluctuations are in general oscillatory and (degenerate cases excluded) do not converge in distribution. This trichotomy is similar to what has been observed in related situations, such as some other branching processes and for Pólya urns. The results lead to a symbolic calculus describing the limits. The asymptotic results also apply to the total of other (random) characteristics of the population.

Asymptotic results for the extinction time of Markov branching processes allowing emigration, I. Random walk decrements

1989 ◽  
Vol 21 (02) ◽  
pp. 243-269 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Population Size ◽  
Large Scale ◽  
Branching Process ◽  
Branching Processes ◽  
Initial Population ◽  
Extinction Time ◽  
Asymptotic Results ◽  
Time To Extinction ◽  
Initial Population Size ◽  
The Mathematical Model

The mathematical model is a Markov branching process which is subjected to catastrophes or large-scale emigration. Catastrophes reduce the population size by independent and identically distributed decrements, and two mechanisms for generating catastrophe epochs are given separate consideration. These are that catastrophes occur at a rate proportional to population size, and as an independent Poisson process. The paper studies some properties of the time to extinction of the modified process in those cases where extinction occurs almost surely. Particular attention is given to limit theorems and the behaviour of the expected extinction time as the initial population size grows. These properties are contrasted with known properties for the case when there is no catastrophe component.

Parametric inference in Markov branching processes with time-dependent random immigration rate

1985 ◽  
Vol 22 (3) ◽  
pp. 503-517 ◽  
Author(s):  
Helmut Pruscha
Keyword(s):  
Point Processes ◽  
Traditional Approach ◽  
Branching Processes ◽  
External Source ◽  
Time Dependent ◽  
Parametric Inference ◽  
Asymptotic Results ◽  
Subcritical Case ◽  
Immigration Rate ◽  
Pearson Type

The present paper deals with continuous-time Markov branching processes allowing immigration. The immigration rate is allowed to be random and time-dependent where randomness may stem from an external source or from state-dependence. Unlike the traditional approach, we base the analysis of these processes on the theory of multivariate point processes. Using the tools of this theory, asymptotic results on parametric inference are derived for the subcritical case. In particular, the limit distributions of some parametric estimators and of Pearson-type statistics for testing simple and composite hypotheses are established.

Multitype infinite-allele branching processes in continuous time

2017 ◽  
Vol 54 (2) ◽  
pp. 550-568
Author(s):  
Thomas O. McDonald ◽  
Marek Kimmel
Keyword(s):  
Tumor Growth ◽  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Clonal Evolution ◽  
Asymptotic Results ◽  
General Process ◽  
Binary Splitting ◽  
Passenger Mutations ◽  
Viral Reproduction

Abstract We introduce extensions to an infinite-allele branching process that allows for multiple types to exist alongside labels. We consider a Markov branching process and general branching process under different assumptions, and show asymptotic results about the growth of the labels as well as the frequency spectrum. These results are motivated by two separate models. The Markov binary splitting results are motivated by a model of clonal evolution in cancer that considers the effect of both driver and passenger mutations on tumor growth. The general process has applications in viral reproduction and dynamics.

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