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The rescaled Pólya urn: local reinforcement and chi-squared goodness-of-fit test

Part of: Parametric inference Limit theorems Markov processes

Published online by Cambridge University Press:  18 October 2022

Giacomo Aletti Open the ORCID record for Giacomo Aletti [Opens in a new window]  and
Irene Crimaldi
Giacomo Aletti*
Affiliation:
Università degli Studi di Milano
Irene Crimaldi*
Affiliation:
IMT School for Advanced Studies Lucca
*
*Postal address: ADAMSS Center, Università degli Studi di Milano, Milan, Italy.
**Postal address: IMT School for Advanced Studies Lucca, Lucca, Italy.
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Abstract

Motivated by recent studies of big samples, this work aims to construct a parametric model which is characterized by the following features: (i) a ‘local’ reinforcement, i.e. a reinforcement mechanism mainly based on the last observations, (ii) a random persistent fluctuation of the predictive mean, and (iii) a long-term almost sure convergence of the empirical mean to a deterministic limit, together with a chi-squared goodness-of-fit result for the limit probabilities. This triple purpose is achieved by the introduction of a new variant of the Eggenberger–Pólya urn, which we call the rescaled Pólya urn. We provide a complete asymptotic characterization of this model, pointing out that, for a certain choice of the parameters, it has properties different from the ones typically exhibited by the other urn models in the literature. Therefore, beyond the possible statistical application, this work could be interesting for those who are concerned with stochastic processes with reinforcement.

Keywords

Empirical meancentral limit theoremchi-squared testcompact Markov chainPólya urnpredictive meanpreferential attachmentreinforcement learningreinforced stochastic processurn model

MSC classification

Primary: 60F05: Central limit and other weak theorems 62F03: Hypothesis testing
Secondary: 60J05: Discrete-time Markov processes on general state spaces 62F05: Asymptotic properties of tests
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 3 , September 2022 , pp. 849 - 879
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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