Convergence to a Continuous State Branching Process

2016 ◽  
pp. 13-21
Author(s):  
Étienne Pardoux
Keyword(s):  
Branching Process ◽  
Continuous State ◽  
Continuous State Branching Process

Distributions of jumps in a continuous-state branching process with immigration

2016 ◽  
Vol 53 (4) ◽  
pp. 1166-1177 ◽  
Author(s):  
Xin He ◽  
Zenghu Li
Keyword(s):  
Branching Process ◽  
Lévy Measure ◽  
Borel Set ◽  
Jump Size ◽  
Distributional Properties ◽  
Continuous State ◽  
Continuous State Branching Process ◽  
Jump Time ◽  
Branching Process With Immigration

Abstract We study the distributional properties of jumps in a continuous-state branching process with immigration. In particular, a representation is given for the distribution of the first jump time of the process with jump size in a given Borel set. From this result we derive a characterization for the distribution of the local maximal jump of the process. The equivalence of this distribution and the total Lévy measure is then studied. For the continuous-state branching process without immigration, we also study similar problems for its global maximal jump.

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.

Skeletal stochastic differential equations for continuous-state branching processes

2019 ◽  
Vol 56 (4) ◽  
pp. 1122-1150 ◽  
Author(s):  
D. Fekete ◽  
J. Fontbona ◽  
A. E. Kyprianou
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Continuous State ◽  
Continuous State Branching Process ◽  
Common Framework ◽  
Watson Process ◽  
Skeletal Representation

AbstractIt is well understood that a supercritical continuous-state branching process (CSBP) is equal in law to a discrete continuous-time Galton–Watson process (the skeleton of prolific individuals) whose edges are dressed in a Poissonian way with immigration which initiates subcritical CSBPs (non-prolific mass). Equally well understood in the setting of CSBPs and superprocesses is the notion of a spine or immortal particle dressed in a Poissonian way with immigration which initiates copies of the original CSBP, which emerges when conditioning the process to survive eternally. In this article we revisit these notions for CSBPs and put them in a common framework using the well-established language of (coupled) stochastic differential equations (SDEs). In this way we are able to deal simultaneously with all types of CSBPs (supercritical, critical, and subcritical) as well as understanding how the skeletal representation becomes, in the sense of weak convergence, a spinal decomposition when conditioning on survival. We have two principal motivations. The first is to prepare the way to expand the SDE approach to the spatial setting of superprocesses, where recent results have increasingly sought the use of skeletal decompositions to transfer results from the branching particle setting to the setting of measure valued processes. The second is to provide a pathwise decomposition of CSBPs in the spirit of genealogical coding of CSBPs via Lévy excursions, albeit precisely where the aforesaid coding fails to work because the underlying CSBP is supercritical.

scholarly journals On Prolific Individuals in a Supercritical Continuous-State Branching Process

2008 ◽  
Vol 45 (03) ◽  
pp. 714-726 ◽  
Author(s):  
Jean Bertoin ◽  
Joaquin Fontbona ◽  
Servet Martínez
Keyword(s):  
Branching Process ◽  
Continuous State ◽  
Continuous State Branching Process ◽  
Infinite Descent

We describe the genealogy of individuals with infinite descent in a supercritical continuous-state branching process.

Sign in / Sign up

Export Citation Format

Share Document