On Pólya–Friedman random walks

Abstract

The Pólya process is an urn scheme arising in the context of contagion spreading. It exhibits unstable persistence effects. The Friedman urn process is dual to the Pólya one with antipersistent stabilizing effects. It appears in a safety campaign problem. A Pólya–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 Pólya–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.