## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 27, Number 5 (2017), 2956-3003.

### Central limit theorem for an adaptive randomly reinforced urn model

Andrea Ghiglietti, Anand N. Vidyashankar, and William F. Rosenberger

#### Abstract

The generalized Pólya urn (GPU) models and their variants have been investigated in several disciplines. However, typical assumptions made with respect to the GPU do not include urn models with a diagonal replacement matrix, which arise in several applications, specifically in clinical trials. To facilitate mathematical analyses of models in these applications, we introduce an adaptive randomly reinforced urn model that uses accruing statistical information to adaptively skew the urn proportion toward specific targets. We study several probabilistic aspects that are important in implementing the urn model in practice. Specifically, we establish the law of large numbers and a central limit theorem for the number of sampled balls. To establish these results, we develop new techniques involving last exit times and crossing time analyses of the proportion of balls in the urn. To obtain precise estimates in these techniques, we establish results on the harmonic moments of the total number of balls in the urn. Finally, we describe our main results in the context of an application to response-adaptive randomization in clinical trials. Our simulation experiments in this context demonstrate the ease and scope of our model.

#### Article information

**Source**

Ann. Appl. Probab., Volume 27, Number 5 (2017), 2956-3003.

**Dates**

Received: February 2015

Revised: October 2016

First available in Project Euclid: 3 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1509696039

**Digital Object Identifier**

doi:10.1214/16-AAP1274

**Mathematical Reviews number (MathSciNet)**

MR3719951

**Zentralblatt MATH identifier**

1379.60025

**Subjects**

Primary: 60F15: Strong theorems 60F05: Central limit and other weak theorems 60E20 60G99: None of the above, but in this section

Secondary: 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) [See also 68W20, 68W40] 97K50: Probability theory

**Keywords**

Clinical trials crossing times harmonic moments last exit times generalized Pólya urn target allocation

#### Citation

Ghiglietti, Andrea; Vidyashankar, Anand N.; Rosenberger, William F. Central limit theorem for an adaptive randomly reinforced urn model. Ann. Appl. Probab. 27 (2017), no. 5, 2956--3003. doi:10.1214/16-AAP1274. https://projecteuclid.org/euclid.aoap/1509696039