The rescaled p?lya urn: local reinforcement and chi-squared goodness-of-fit test

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 per-sistent 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-Polya urn, which we call the rescaled Polya 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 exhib-ited 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.