Stationary distributions for a class of generalized Fleming–Viot processes

Handa, Kenji

doi:10.1214/12-aop829

Cited by 4 publications

(12 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Exact calculations are always likely to be difficult because of the jump process nature of the Λ-Fleming-Viot process. A first step in this direction, for certain classes of Fleming-Viot processes where stationary distributions are characterized, can be found in Handa (2012).…”

Section: Introductionmentioning

confidence: 99%

The Λ-Fleming-Viot Process and a Connection with Wright-Fisher Diffusion

Griffiths

2014

Adv. Appl. Probab.

View full text Add to dashboard Cite

The d-dimensional Λ-Fleming-Viot generator acting on functions g(x x x), with x x x being a vector of d allele frequencies, can be written as a Wright-Fisher generator acting on functions g with a modified random linear argument of x x x induced by partitioning occurring in the Λ-Fleming-Viot process. The eigenvalues and right polynomial eigenvectors are easy to see from the representation. The two-dimensional process, which has a one-dimensional generator, is considered in detail. A nonlinear equation is found for the Green's function. In a model with genic selection the fixation probability can be computed from an algorithm in the paper.An application in the infinitely-many-alleles Λ-Fleming-Viot process is finding an interesting identity for the frequency spectrum of alleles that is based on sizebiassing.The moment dual process in the Fleming-Viot process is the usual Λ-coalescent tree back in time. In the Wright-Fisher representation using a different set of polynomials g n (x) as test functions produces a dual death process which has a similarity to the Kingman coalescent and decreases by units of one. The eigenvalues of the process are analogous to the Jacobi polynomials when expressed in terms of g n (x), playing the role of x n .

show abstract

Section: Introductionmentioning

confidence: 99%

The Λ-Fleming-Viot Process and a Connection with Wright-Fisher Diffusion

Griffiths

2014

Adv. Appl. Probab.

View full text Add to dashboard Cite

show abstract

“…. , r n ) = f 1 (r 1 ) · · · f n (r n ) with f i ∈ C + (E) is dense in C(E n ), one can show, with the help of the expression (3.2) in [4] for A α Φ f with f ∈ C(E n ), that (5.7) extends to any Φ ∈ F 1 . Indeed, that expression takes the form…”

Section: An Application To Generalized Fleming-viot Processesmentioning

confidence: 99%

“…, η, f n ) ∈ R n f for any η ∈ M(E) • . Intuitively, these conditions enable one to control the effect of long-range jumps governed by stable laws, and are inspired by the calculations in the proof of Proposition 3.4 in [4].…”

Section: The Measure-valued α-Cir Modelsmentioning

confidence: 99%

“…Notice that in one dimension (5.2) takes the form (1.3). Under the assumption that m(E) > 1, it was proved in Proposition 3.4 of [4] that…”

Section: An Application To Generalized Fleming-viot Processesmentioning

confidence: 99%

“…where G is any smooth function on [0, 1], F is defined by F (z 1 , z 2 ) = G(z 1 /(z 1 + z 2 )), C is a positive constant independent of G and A α is the generator of a jump-type version of the Wright-Fisher diffusion model. (See (1.3) in [4] for a concrete expression for A α or (5.1) below for its generalization.) A significant consequence of (1.3) is that Dirichlet form associated with A α is, up to some multiplicative constant, a restriction of Dirichlet form associated with the two independent α-CIR models.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Ergodic properties for $\alpha$-CIR models and a class of generalized Fleming-Viot processes

Handa

2014

Electron. J. Probab.

Self Cite

View full text Add to dashboard Cite

Dedicated to Professor Ken-iti Sato on the occasion of his 80th birthdayWe discuss a Markov jump process regarded as a variant of the CIR (Cox-Ingersoll-Ross) model and its infinite-dimensional extension. These models belong to a class of measure-valued branching processes with immigration, whose jump mechanisms are governed by certain stable laws. The main result gives a lower spectral gap estimate for the generator. As an application, a certain ergodic property is shown for the generalized Fleming-Viot process obtained as the time-changed ratio process.1 AMS 2010 subject classifications. Primary 60J75; secondary 60G57.

show abstract

Stable continuous-state branching processes with immigration and Beta-Fleming-Viot processes with immigration

Foucart

Hénard

2013

Electron. J. Probab.

View full text Add to dashboard Cite

Branching processes and Fleming-Viot processes are two main models in stochastic population theory. Incorporating an immigration in both models, we generalize the results of Shiga (1990) and Birkner et al. (2005) which respectively connect the Feller diffusion with the classical Fleming-Viot process and the α-stable continuous state branching process with the Beta(2 − α, α)-generalized Fleming-Viot process. In a recent work, a new class of probabilitymeasure valued processes, called M -generalized Fleming-Viot processes with immigration, has been set up in duality with the so-called M -coalescents. The purpose of this article is to investigate the links between this new class of processes and the continuous-state branching processes with immigration. In the specific case of the α-stable branching process conditioned to be never extinct, we get that its genealogy is given, up to a random time change, by a Beta(2 − α, α − 1)coalescent.

show abstract

Stationary distributions for a class of generalized Fleming–Viot processes

Cited by 4 publications

References 24 publications

The Λ-Fleming-Viot Process and a Connection with Wright-Fisher Diffusion

The Λ-Fleming-Viot Process and a Connection with Wright-Fisher Diffusion

Ergodic properties for $\alpha$-CIR models and a class of generalized Fleming-Viot processes

Stable continuous-state branching processes with immigration and Beta-Fleming-Viot processes with immigration

Contact Info

Product

Resources

About