Open Access
August 2018 Dynamics of an adaptive randomly reinforced urn
Giacomo Aletti, Andrea Ghiglietti, Anand N. Vidyashankar
Bernoulli 24(3): 2204-2255 (August 2018). DOI: 10.3150/17-BEJ926


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.


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Giacomo Aletti. Andrea Ghiglietti. Anand N. Vidyashankar. "Dynamics of an adaptive randomly reinforced urn." Bernoulli 24 (3) 2204 - 2255, August 2018.


Received: 1 August 2015; Revised: 1 February 2017; Published: August 2018
First available in Project Euclid: 2 February 2018

zbMATH: 06839265
MathSciNet: MR3757528
Digital Object Identifier: 10.3150/17-BEJ926

Keywords: central limit theorems , crossing times , generalized Pólya urn , harmonic moments , Reinforced processes , strong and weak consistency

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 3 • August 2018
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