On Polya-Friedman random walks
Résumé
The P\'{o}lya process is an urn scheme arising in the context of the contagion spreading. It exhibits unstable persistence effects. The Friedman urn process is dual to the P\'{o}lya one with antipersistent stabilizing effects. It appears in a safety campaign problem. A P\'{o}lya-Friedman urn process is investigated with a tuning persistence parameter extrapolating the latter two extreme processes. The study includes the diffusion approximations of both the P\'{o}lya-Friedman proportion process and the population gap random walk. The structure of the former is a generalized Wright-Fisher diffusion appearing in Population Genetics. The correlation structure of the latter presents an anomalous character at a critical value of the persistence parameter.
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