## Abstract

The Pitman-Yor process is a random discrete measure. The random weights or masses follow the two-parameter Poisson-Dirichlet distribution with parameters 0 < α < 1, θ >-α. The parameters α and θ correspond to the stable and gamma components, respectively. The distribution of atoms is given by a probability η. In this paper we consider the limit theorems for the Pitman-Yor process and the two-parameter Poisson-Dirichlet distribution. These include the law of large numbers, fluctuations, and moderate or large deviation principles. The limiting procedures involve either α tending to 0 or 1. They arise naturally in genetics and physics such as the asymptotic coalescence time for explosive branching process and the approximation to the generalized random energy model for disordered systems.

Original language | English |
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Pages (from-to) | 581-602 |

Number of pages | 22 |

Journal | Advances in Applied Probability |

Volume | 49 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Jun 2017 |

Externally published | Yes |

## Keywords

- Poisson-Dirichlet distribution
- Yor process
- explosive branching
- large deviation
- phase transition
- random energy model