Some properties of continuous-state branching processes, with applications to Bartoszyński’s virus model

1985 ◽  
Vol 17 (1) ◽  
pp. 23-41 ◽  
Author(s):  
Anthony G. Pakes ◽  
A. C. Trajstman
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Compound Poisson Process ◽  
Explicit Construction ◽  
General Context ◽  
Compound Poisson ◽  
Continuous State ◽  
Virus Model

It is known that Bartoszyński’s model for the growth of rabies virus in an infected host is a continuous branching process. We show by explicit construction that any such process is a randomly time-transformed compound Poisson process having a negative linear drift.This connection is exploited to obtain limit theorems for the population size and for the jump times in the rabies model. Some of these results are obtained in a more general context wherein the compound Poisson process is replaced by a subordinator.

Some properties of continuous-state branching processes, with applications to Bartoszyński’s virus model

1985 ◽  
Vol 17 (01) ◽  
pp. 23-41
Author(s):  
Anthony G. Pakes ◽  
A. C. Trajstman
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Compound Poisson Process ◽  
Explicit Construction ◽  
General Context ◽  
Compound Poisson ◽  
Continuous State ◽  
Virus Model

It is known that Bartoszyński’s model for the growth of rabies virus in an infected host is a continuous branching process. We show by explicit construction that any such process is a randomly time-transformed compound Poisson process having a negative linear drift. This connection is exploited to obtain limit theorems for the population size and for the jump times in the rabies model. Some of these results are obtained in a more general context wherein the compound Poisson process is replaced by a subordinator.

scholarly journals Continuous branching processes : the discrete hidden in the continuous : Dedicated to Claude Lobry

2008 ◽  
Vol Volume 9, 2007 Conference in... ◽  
Author(s):  
Etienne Pardoux
Keyword(s):  
Poisson Process ◽  
Branching Process ◽  
Branching Processes ◽  
Compound Poisson Process ◽  
Levy Process ◽  
Initial Condition ◽  
Compound Poisson ◽  
International Audience ◽  
Number Of Individuals ◽  
Branching Property

International audience Feller diffusion is a continuous branching process. The branching property tells us that for t > 0 fixed, when indexed by the initial condition, it is a subordinator (i. e. a positive–valued Lévy process), which is fact is a compound Poisson process. The number of points of this Poisson process can be interpreted as the number of individuals whose progeny survives during a number of generations of the order of t × N, where N denotes the size of the population, in the limit N ―>µ. This fact follows from recent results of Bertoin, Fontbona, Martinez [1]. We compare them with older results of de O’Connell [7] and [8]. We believe that this comparison is useful for better understanding these results. There is no new result in this presentation. La diffusion de Feller est un processus de branchement continu. La propriété de branchement nous dit que à t > 0 fixé, indexé par la condition initiale, ce processus est un subordinateur (processus de Lévy à valeurs positives), qui est en fait un processus de Poisson composé. Le nombre de points de ce processus de Poisson s’interprète comme le nombre d’individus dont la descendance survit au cours d’un nombre de générations de l’ordre de t × N, où N désigne la taille de la population, dans la limite N --> µ. Ce fait découle de résultats récents de Bertoin, Fontbona, Martinez [1]. Nous le rapprochons de résultats plus anciens de O’Connell [7] et [8]. Ce rapprochement nous semble aider à mieux comprendre ces résultats. Cet article ne contient pas de résultat nouveau.

scholarly journals Some limit theorems for continuous-state branching processes

1988 ◽  
Vol 44 (1) ◽  
pp. 71-87 ◽  
Author(s):  
Anthony G Pakes
Keyword(s):  
Continuous Time ◽  
Limit Theorems ◽  
Branching Processes ◽  
Random Time ◽  
Time Transformation ◽  
Compound Poisson ◽  
Linear Drift ◽  
Continuous State ◽  
Drift Parameter ◽  
Random Variation

AbstractThe most general continuous time and state branching (C.B.) process (Xt) can be constructed as a certain random time transformation of a spectrally positive Levy process. When the generating process is compound Poisson with a superimposed negative linear drift and the C.B. process is not supercritical, then there is a random time T such that Xt+T = e-ctXT where c > 0 is the drift parameter. Thus T is the last epoch of random variation.

On the subcritical Bellman-Harris process with immigration

1974 ◽  
Vol 11 (4) ◽  
pp. 652-668 ◽  
Author(s):  
A. G. Pakes ◽  
Norman Kaplan
Keyword(s):  
Poisson Process ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Compound Poisson Process ◽  
Limiting Distribution ◽  
Batch Arrival ◽  
Compound Poisson ◽  
Queueing Process ◽  
Age Dependent ◽  
Necessary And Sufficient

Some necessary and sufficient conditions are found for the existence of a proper non-degenerate limiting distribution for a Bellman-Harris age-dependent branching process with a compound renewal immigration component. A number of these results are applicable to the batch arrival GI/G/∞ queueing process. Some aspects of the situation when there is no such limiting distribution are considered. The situation when the immigration component is a nonhomogeneous compound Poisson process is briefly considered.

scholarly journals Limit theorems for the fractional nonhomogeneous Poisson process

2019 ◽  
Vol 56 (01) ◽  
pp. 246-264 ◽  
Author(s):  
Nikolai Leonenko ◽  
Enrico Scalas ◽  
Mailan Trinh
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Poisson Process ◽  
Limit Theorems ◽  
Compound Poisson Process ◽  
Nonhomogeneous Poisson Process ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Compound Poisson ◽  
Finite Dimensional

AbstractThe fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverse α-stable subordinator. We propose a similar definition for the (nonhomogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.

On the subcritical Bellman-Harris process with immigration

1974 ◽  
Vol 11 (04) ◽  
pp. 652-668 ◽  
Author(s):  
A. G. Pakes ◽  
Norman Kaplan
Keyword(s):  
Poisson Process ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Compound Poisson Process ◽  
Limiting Distribution ◽  
Batch Arrival ◽  
Compound Poisson ◽  
Queueing Process ◽  
Age Dependent ◽  
Necessary And Sufficient

Some necessary and sufficient conditions are found for the existence of a proper non-degenerate limiting distribution for a Bellman-Harris age-dependent branching process with a compound renewal immigration component. A number of these results are applicable to the batch arrival GI/G/∞ queueing process. Some aspects of the situation when there is no such limiting distribution are considered. The situation when the immigration component is a nonhomogeneous compound Poisson process is briefly considered.

scholarly journals Skeletal stochastic differential equations for superprocesses

2020 ◽  
Vol 57 (4) ◽  
pp. 1111-1134
Author(s):  
Dorottya Fekete ◽  
Joaquin Fontbona ◽  
Andreas E. Kyprianou
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Subcritical Case ◽  
Markov Branching Process ◽  
Continuous State ◽  
Common Framework ◽  
Infinite Line ◽  
Current Article ◽  
Lines Of Descent ◽  
Skeletal Representation

AbstractIt is well understood that a supercritical superprocess is equal in law to a discrete Markov branching process whose genealogy is dressed in a Poissonian way with immigration which initiates subcritical superprocesses. The Markov branching process corresponds to the genealogical description of prolific individuals, that is, individuals who produce eternal genealogical lines of descent, and is often referred to as the skeleton or backbone of the original superprocess. The Poissonian dressing along the skeleton may be considered to be the remaining non-prolific genealogical mass in the superprocess. Such skeletal decompositions are equally well understood for continuous-state branching processes (CSBP).In a previous article [16] we developed an SDE approach to study the skeletal representation of CSBPs, which provided a common framework for the skeletal decompositions of supercritical and (sub)critical CSBPs. It also helped us to understand how the skeleton thins down onto one infinite line of descent when conditioning on survival until larger and larger times, and eventually forever.Here our main motivation is to show the robustness of the SDE approach by expanding it to the spatial setting of superprocesses. The current article only considers supercritical superprocesses, leaving the subcritical case open.

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