### Abstract

Language | English |
---|---|

Pages | 254-281 |

Journal | Journal of Applied Probability |

Volume | 55 |

Issue number | 1 |

DOIs | |

Status | Published - 1 Mar 2018 |

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*Journal of Applied Probability*,

*55*(1), 254-281. https://doi.org/10.1017/jpr.2018.16

**Multiple drawing multi-colour urns by stochastic approximation.** / Lasmar, Nabil; Mailler, Cecile; Selmi, Olfa.

Research output: Contribution to journal › Article

*Journal of Applied Probability*, vol. 55, no. 1, pp. 254-281. https://doi.org/10.1017/jpr.2018.16

}

TY - JOUR

T1 - Multiple drawing multi-colour urns by stochastic approximation

AU - Lasmar, Nabil

AU - Mailler, Cecile

AU - Selmi, Olfa

PY - 2018/3/1

Y1 - 2018/3/1

N2 - A classical P\'olya urn scheme is a Markov process whose evolution is encoded by a replacement matrix $(R_{i,j})_{1\leq i,j\leq d}$. At every discrete time-step, we draw a ball uniformly at random, denote its colour $c$, and replace it in the urn together with $R_{c,j}$ balls of colour $j$ (for all $1\leq j\leq d$).We study multi-drawing P\'olya 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 by Kuba \& Mahmoud (ArXiv:1503.09069 and 1509.09053).These authors prove second order asymptotic results in the $2$-colour case, under the so-called {\it balance} and {\it 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 \& Mahmoud, but without the artificial {\it affinity} hypothesis, and, for the first time in the literature, in the $d$-colour case ($d\geq 3$). We also give some partial results in the two-colour non-balanced case, the novelty here being that the only results for this case currently in the literature are for particular examples.

AB - A classical P\'olya urn scheme is a Markov process whose evolution is encoded by a replacement matrix $(R_{i,j})_{1\leq i,j\leq d}$. At every discrete time-step, we draw a ball uniformly at random, denote its colour $c$, and replace it in the urn together with $R_{c,j}$ balls of colour $j$ (for all $1\leq j\leq d$).We study multi-drawing P\'olya 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 by Kuba \& Mahmoud (ArXiv:1503.09069 and 1509.09053).These authors prove second order asymptotic results in the $2$-colour case, under the so-called {\it balance} and {\it 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 \& Mahmoud, but without the artificial {\it affinity} hypothesis, and, for the first time in the literature, in the $d$-colour case ($d\geq 3$). We also give some partial results in the two-colour non-balanced case, the novelty here being that the only results for this case currently in the literature are for particular examples.

U2 - 10.1017/jpr.2018.16

DO - 10.1017/jpr.2018.16

M3 - Article

VL - 55

SP - 254

EP - 281

JO - Journal of Applied Probability

T2 - Journal of Applied Probability

JF - Journal of Applied Probability

SN - 0021-9002

IS - 1

ER -