  Motivated by recent studies of big samples, this work aims at constructing 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 persistent fluctuation of the predictive mean, and
(iii) a long-term convergence of the empirical mean to a deterministic limit,
together with a chi-squared goodness of fit result. This triple purpose has
been achieved by the introduction of a new variant of the Eggenberger-Polya
urn, that 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 exhibited from
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.
