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Multiple drawing multi-colour urns by stochastic approximation

Part of: Limit theorems Sequential methods

Published online by Cambridge University Press:  28 March 2018

Nabil Lasmar ,
Cécile Mailler  and
Olfa Selmi
Nabil Lasmar*
Affiliation:
Institut Préparatoire aux Études d'Ingénieur
Cécile Mailler*
Affiliation:
University of Bath
Olfa Selmi*
Affiliation:
Faculté des Sciences de Monastir
*
* Postal address: Département des Mathématiques, Institut Préparatoire aux Études d'Ingénieur, Monastir, Tunisia.
** Postal address: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK. Email address: c.mailler@bath.ac.uk
*** Postal address: Département des Mathématiques, Faculté des Sciences de Monastir, Monastir, Tunisia.
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Abstract

A classical Pólya urn scheme is a Markov process where the evolution is encoded by a replacement matrix (Ri, j)1 ≤ i, j ≤ d. At every discrete time-step, we draw a ball uniformly at random, denote its colour c, and replace it in the urn together with Rc, j balls of colour j (for all 1 ≤ j ≤ d). We study multiple drawing Pólya urns, where the replacement rule depends on the random drawing of a set of m balls from the urn (with or without replacement). Many particular examples of this situation have been studied in the literature, but the only general results are due to Kuba and Mahmoud (2017). These authors proved second-order asymptotic results in the two-colour case, under the so-called balance and affinity assumptions, the latter being somewhat artificial. The main idea of this work is to apply stochastic approximation methods to this problem, which enables us to prove analogous results to Kuba and Mahmoud, but without the artificial affinity hypothesis, and, for the first time in the literature, in the d-colour case (d ≥ 3). We also provide some partial results in the two-colour nonbalanced case, the novelty here being that the only results for this case currently in the literature are for particular examples.

Keywords

Multiple drawing Pólya urnlimit theoremstochastic approximationreinforced processdiscrete-time martingale

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60F15: Strong theorems 62L20: Stochastic approximation
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018 , pp. 254 - 281
Copyright
Copyright © Applied Probability Trust 2018 

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