scholarly journals Total length of the genealogical tree for quadratic stationary continuous-state branching processes

2016 ◽  
Vol 52 (3) ◽  
pp. 1321-1350 ◽  
Author(s):  
Hongwei Bi ◽  
Jean-François Delmas
Keyword(s):  
Branching Processes ◽  
Genealogical Tree ◽  
Total Length ◽  
Continuous State

scholarly journals Exact simulation of the genealogical tree for a stationary branching population and application to the asymptotics of its total length

2021 ◽  
Vol 53 (2) ◽  
pp. 537-574
Author(s):  
Romain Abraham ◽  
Jean-François Delmas
Keyword(s):  
Fixed Time ◽  
Genealogical Tree ◽  
Total Length ◽  
Constant Size ◽  
Continuous State ◽  
Continuous State Branching Process ◽  
Size Population ◽  
Quadratic Branching ◽  
Branching Process With Immigration ◽  
Extant Population

AbstractWe consider a model of a stationary population with random size given by a continuous-state branching process with immigration with a quadratic branching mechanism. We give an exact elementary simulation procedure for the genealogical tree of n individuals randomly chosen among the extant population at a given time. Then we prove the convergence of the renormalized total length of this genealogical tree as n goes to infinity; see also Pfaffelhuber, Wakolbinger and Weisshaupt (2011) in the context of a constant-size population. The limit appears already in Bi and Delmas (2016) but with a different approximation of the full genealogical tree. The proof is based on the ancestral process of the extant population at a fixed time, which was defined by Aldous and Popovic (2005) in the critical case.

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.

scholarly journals Backbone Decomposition of Multitype Superprocesses

2021 ◽  
Author(s):  
Dorottya Fekete ◽  
Sandra Palau ◽  
Juan Carlos Pardo ◽  
Jose Luis Pérez
Keyword(s):  
Branching Processes ◽  
Easy Consequence ◽  
Continuous State ◽  
The One

AbstractIn this paper, we provide a construction of the so-called backbone decomposition for multitype supercritical superprocesses. While backbone decompositions are fairly well known for both continuous-state branching processes and superprocesses in the one-type case, so far no such decompositions or even description of prolific genealogies have been given for the multitype cases. Here we focus on superprocesses, but by turning the movement off, we get the prolific backbone decomposition for multitype continuous-state branching processes as an easy consequence of our results.

scholarly journals Proof(s) of the Lamperti representation of continuous-state branching processes

2009 ◽  
Vol 6 (0) ◽  
pp. 62-89 ◽  
Author(s):  
Ma. Emilia Caballero ◽  
Amaury Lambert ◽  
Gerónimo Uribe Bravo
Keyword(s):  
Branching Processes ◽  
Continuous State

scholarly journals Continuous-State Branching Processes and Self-Similarity

2008 ◽  
Vol 45 (04) ◽  
pp. 1140-1160 ◽  
Author(s):  
A. E. Kyprianou ◽  
J. C. Pardo
Keyword(s):  
Markov Processes ◽  
Branching Processes ◽  
Self Similarity ◽  
Lamperti Transformation ◽  
Continuous State ◽  
Self Similar

In this paper we study the α-stable continuous-state branching processes (for α ∈ (1, 2]) and the α-stable continuous-state branching processes conditioned never to become extinct in the light of positive self-similarity. Understanding the interaction of the Lamperti transformation for continuous-state branching processes and the Lamperti transformation for positive, self-similar Markov processes gives access to a number of explicit results concerning the paths of α-stable continuous-state branching processes and α-stable continuous-state branching processes conditioned never to become extinct.

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