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.

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 Species Dynamics in the Two-Parameter Poisson-Dirichlet Diffusion Model

2014 ◽  
Vol 51 (01) ◽  
pp. 174-190 ◽  
Author(s):  
Matteo Ruggiero
Keyword(s):  
Mathematical Properties ◽  
State Dependent ◽  
Continuous State ◽  
Continuous State Branching Process ◽  
Biological Interpretation ◽  
Species Dynamics ◽  
The One ◽  
Branching Process With Immigration ◽  
Parameter Case ◽  
Two Parameter

The recently introduced two-parameter infinitely-many-neutral-alleles model extends the celebrated one-parameter version (which is related to Kingman's distribution) to diffusive two-parameter Poisson-Dirichlet frequencies. In this paper we investigate the dynamics driving the species heterogeneity underlying the two-parameter model. First we show that a suitable normalization of the number of species is driven by a critical continuous-state branching process with immigration. Secondly, we provide a finite-dimensional construction of the two-parameter model, obtained by means of a sequence of Feller diffusions of Wright-Fisher flavor which feature finitely many types and inhomogeneous mutation rates. Both results provide insight into the mathematical properties and biological interpretation of the two-parameter model, showing that it is structurally different from the one-parameter case in that the frequency dynamics are driven by state-dependent rather than constant quantities.

scholarly journals Species Dynamics in the Two-Parameter Poisson-Dirichlet Diffusion Model

2014 ◽  
Vol 51 (1) ◽  
pp. 174-190 ◽  
Author(s):  
Matteo Ruggiero
Keyword(s):  
Mathematical Properties ◽  
State Dependent ◽  
Continuous State ◽  
Continuous State Branching Process ◽  
Biological Interpretation ◽  
Species Dynamics ◽  
The One ◽  
Branching Process With Immigration ◽  
Parameter Case ◽  
Two Parameter

The recently introduced two-parameter infinitely-many-neutral-alleles model extends the celebrated one-parameter version (which is related to Kingman's distribution) to diffusive two-parameter Poisson-Dirichlet frequencies. In this paper we investigate the dynamics driving the species heterogeneity underlying the two-parameter model. First we show that a suitable normalization of the number of species is driven by a critical continuous-state branching process with immigration. Secondly, we provide a finite-dimensional construction of the two-parameter model, obtained by means of a sequence of Feller diffusions of Wright-Fisher flavor which feature finitely many types and inhomogeneous mutation rates. Both results provide insight into the mathematical properties and biological interpretation of the two-parameter model, showing that it is structurally different from the one-parameter case in that the frequency dynamics are driven by state-dependent rather than constant quantities.

scholarly journals Asymptotic behavior of critical, irreducible multi-type continuous state and continuous time branching processes with immigration

2016 ◽  
Vol 16 (04) ◽  
pp. 1650008 ◽  
Author(s):  
Mátyás Barczy ◽  
Gyula Pap
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Bessel Process ◽  
Step Functions ◽  
Squared Bessel Process ◽  
Continuous State ◽  
Random Step ◽  
Branching Process With Immigration ◽  
Continuous Time Branching Processes

Under natural assumptions, a Feller type diffusion approximation is derived for critical, irreducible multi-type continuous state and continuous time branching processes with immigration. Namely, it is proved that a sequence of appropriately scaled random step functions formed from a critical, irreducible multi-type continuous state and continuous time branching process with immigration converges weakly towards a squared Bessel process supported by a ray determined by the Perron vector of a matrix related to the branching mechanism of the branching process in question.

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.

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