Urns with simultaneous drawing

Abstract

In classical urn models, one usually draws one ball with replacement at each time unit and then adds one ball of the same colour. Given a weight sequence $(w_k)_{k\in\N}$, the probability of drawing a ball of a certain colour is proportional to $w_k$ where $k$ is the number of balls of this colour. A classical result states that an urn fixates on one colour after a finite time if an only if $\sum_{0}^\infty w_k^{-1} < \infty$. In this paper we shall study the case when at each time unit we draw with replacement a number $d\in\N$ of balls and then add $d$ new balls of matching colours. The main goal is to prove that the result in the case of maximal interaction generalizes assuming in addition that $(w_k)_{k\in\N}$ is non-decreasing.