Stochastic approximations to the Pitman-Yor process

TL;DR

This paper introduces a novel truncation method for the Pitman-Yor process using a random stopping rule, providing theoretical error control and a practical sampling algorithm for approximating its functionals.

Contribution

It presents a new stochastic truncation approach for the Pitman-Yor process with proven error bounds and an efficient sampling algorithm for practical applications.

Findings

Asymptotic distribution of the truncation point derived

Error control in total variation distance established

Sampling algorithm demonstrated effective approximation

Abstract

In this paper we consider approximations to the popular Pitman-Yor process obtained by truncating the stick-breaking representation. The truncation is determined by a random stopping rule that achieves an almost sure control on the approximation error in total variation distance. We derive the asymptotic distribution of the random truncation point as the approximation error epsilon goes to zero in terms of a polynomially tilted positive stable distribution. The practical usefulness and effectiveness of this theoretical result is demonstrated by devising a sampling algorithm to approximate functionals of the epsilon-version of the Pitman-Yor process.