## The Annals of Applied Probability

### Alpha-diversity processes and normalized inverse-Gaussian diffusions

#### Abstract

The infinitely-many-neutral-alleles model has recently been extended to a class of diffusion processes associated with Gibbs partitions of two-parameter Poisson–Dirichlet type. This paper introduces a family of infinite-dimensional diffusions associated with a different subclass of Gibbs partitions, induced by normalized inverse-Gaussian random probability measures. Such diffusions describe the evolution of the frequencies of infinitely-many types together with the dynamics of the time-varying mutation rate, which is driven by an $\alpha$-diversity diffusion. Constructed as a dynamic version, relative to this framework, of the corresponding notion for Gibbs partitions, the latter is explicitly derived from an underlying population model and shown to coincide, in a special case, with the diffusion approximation of a critical Galton–Watson branching process. The class of infinite-dimensional processes is characterized in terms of its infinitesimal generator on an appropriate domain, and shown to be the limit in distribution of a certain sequence of Feller diffusions with finitely-many types. Moreover, a discrete representation is provided by means of appropriately transformed Moran-type particle processes, where the particles are samples from a normalized inverse-Gaussian random probability measure. The relationship between the limit diffusion and the two-parameter model is also discussed.

#### Article information

Source
Ann. Appl. Probab., Volume 23, Number 1 (2013), 386-425.

Dates
First available in Project Euclid: 25 January 2013

https://projecteuclid.org/euclid.aoap/1359124390

Digital Object Identifier
doi:10.1214/12-AAP846

Mathematical Reviews number (MathSciNet)
MR3059239

Zentralblatt MATH identifier
1277.60129

#### Citation

Ruggiero, Matteo; Walker, Stephen G.; Favaro, Stefano. Alpha-diversity processes and normalized inverse-Gaussian diffusions. Ann. Appl. Probab. 23 (2013), no. 1, 386--425. doi:10.1214/12-AAP846. https://projecteuclid.org/euclid.aoap/1359124390

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