## Bayesian Analysis

### Stochastic Approximations to the Pitman–Yor Process

#### 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 random variable. 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.

#### Article information

Source
Bayesian Anal., Advance publication (2018), 19 pages.

Dates
First available in Project Euclid: 30 October 2018

https://projecteuclid.org/euclid.ba/1540865701

Digital Object Identifier
doi:10.1214/18-BA1127

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures

#### Citation

Arbel, Julyan; De Blasi, Pierpaolo; Prünster, Igor. Stochastic Approximations to the Pitman–Yor Process. Bayesian Anal., advance publication, 30 October 2018. doi:10.1214/18-BA1127. https://projecteuclid.org/euclid.ba/1540865701

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#### Supplemental materials

• Supplementary Material of “Stochastic Approximations to the Pitman–Yor Process”. ALGORITHM 3 for generating from a polynomially tilted positive stable random variable (in a separate document).