ALPHA-DIVERSITY PROCESSES AND NORMALIZED INVERSE-GAUSSIAN DIFFUSIONS

作者：Ruggiero Matteo^*; Walker Stephen G; Favaro Stefano

来源：Annals of Applied Probability, 2013, 23(1): 386-425.

DOI：10.1214/12-AAP846

• Gibbs partitions
• Poisson-Dirichlet
• generalized gamma
• infinitely-many-neutral-alleles model
• time-varying mutation rate

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 a-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.

• 出版日期2013-2