## Bernoulli

- Bernoulli
- Volume 24, Number 3 (2018), 2204-2255.

### Dynamics of an adaptive randomly reinforced urn

Giacomo Aletti, Andrea Ghiglietti, and Anand N. Vidyashankar

#### Abstract

Adaptive randomly reinforced urn (ARRU) is a two-color urn model where the updating process is defined by a sequence of non-negative random vectors $\{(D_{1,n},D_{2,n});n\geq1\}$ and randomly evolving thresholds which utilize accruing statistical information for the updates. Let $m_{1}=E[D_{1,n}]$ and $m_{2}=E[D_{2,n}]$. In this paper, we undertake a detailed study of the dynamics of the ARRU model. First, for the case $m_{1}\neq m_{2}$, we establish $L_{1}$ bounds on the increments of the urn proportion, that is, the proportion of ball colors in the urn, at fixed and increasing times under very weak assumptions on the random threshold sequences. As a consequence, we deduce weak consistency of the evolving urn proportions. Second, under slightly stronger conditions, we establish the strong consistency of the urn proportions for all finite values of $m_{1}$ and $m_{2}$. Specifically, we show that when $m_{1}=m_{2}$, the proportion converges to a non-degenerate random variable. Third, we establish the asymptotic distribution, after appropriate centering and scaling, for the proportion of sampled ball colors and urn proportions for the case $m_{1}=m_{2}$. In the process, we resolve the issue concerning the asymptotic distribution of the proportion of sampled ball colors for a randomly reinforced urn (RRU). To address the technical issues, we establish results on the harmonic moments of the total number of balls in the urn at different times under very weak conditions, which is of independent interest.

#### Article information

**Source**

Bernoulli, Volume 24, Number 3 (2018), 2204-2255.

**Dates**

Received: August 2015

Revised: February 2017

First available in Project Euclid: 2 February 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1517540473

**Digital Object Identifier**

doi:10.3150/17-BEJ926

**Mathematical Reviews number (MathSciNet)**

MR3757528

**Zentralblatt MATH identifier**

06839265

**Keywords**

central limit theorems crossing times generalized Pólya urn harmonic moments reinforced processes strong and weak consistency

#### Citation

Aletti, Giacomo; Ghiglietti, Andrea; Vidyashankar, Anand N. Dynamics of an adaptive randomly reinforced urn. Bernoulli 24 (2018), no. 3, 2204--2255. doi:10.3150/17-BEJ926. https://projecteuclid.org/euclid.bj/1517540473