The Rescaled Polya Urn and the Wright-Fisher process with mutation

TL;DR

This paper demonstrates that the multidimensional Wright-Fisher diffusion with mutation can be derived as a limit of rescaled Polya urn models, linking discrete reinforcement processes to continuous diffusion models.

Contribution

It establishes a novel connection between the rescaled Polya urn and the Wright-Fisher diffusion with mutation, extending previous work on urn models.

Findings

Rescaled Polya urns converge to Wright-Fisher diffusion with mutation.

The model exhibits local reinforcement and persistent fluctuations.

Empirical evidence supports the theoretical connection.

Abstract

In [arXiv:1906.10951 (forthcoming on Advances in Applied Probability),arXiv:2011.05933 (published on PLOS ONE)] the authors introduce, study and apply a new variant of the Eggenberger-Polya urn, called the "Rescaled" Polya urn, which, for a suitable choice of the model parameters, 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 almost sure convergence of the empirical mean to a deterministic limit, together with a chi-squared goodness of fit result for the limit probabilities. In this work, motivated by some empirical evidences in [arXiv:2011.05933 (published on PLOS ONE)], we show that the multidimensional Wright-Fisher diffusion with mutation can be obtained as a suitable limit of the predictive means associated to a family of rescaled Polya urns