Urn models with two types of strategies

Abstract

We study an urn process containing red and blue balls and two different strategies to reinforce the urn. Namely, a generalized Pólya-type strategy versus an i.i.d. one. At each step, one of the two reinforcement strategies is chosen by flipping a coin. We study the asymptotic behaviour of this urn model, and prove a law of large numbers, a central limit theorem and a functional limit theorem for the proportion of balls into the urn. A phase transition is also stated.

Non-Markovian random walks with memory lapses

“…For instance, if β i = 1/λ i for all i, we do not get the convergence (3.3). A more accurated discussion about the relation between these parameters and its consequences is provided at [10] in the context of Generalized Pólya-type urns.…”

“…The formulation was given as a generalization of the Polya urn, that is, if β = 1 the conditional probabilities (2.4) define a Polya urn process. In this sense, in González-Navarrete and Lambert [10] it was introduced an urn process with the property of memory lapses. The relation between dependent Bernoulli sequences and Pólya-type urn processes is discussed therein.…”

A hypernetwork-based urn model for explaining collective dynamics

“…At each step, a ball is inspected randomly from the urn and a ball with the same color is added into it [56]. Since the Po ´lya urn offers a simple framework for modeling, some variants of the urn model have been subsequently proposed [57][58][59], e.g., more balls are drawn or added at each step, more colors of balls are included, and more complex dynamic strategies are applied. In our model, we follow the basic ideas of the Po ´lya urn model.…”