P\'olya urn with memory kernel and asymptotic behaviours of autocorrelation function

TL;DR

This paper studies a generalized Pólya urn process with memory kernels, revealing how different decay rates of memory influence the process's autocorrelation behavior and phase transitions.

Contribution

It introduces arbitrary memory kernels into the Pólya urn model and analyzes their impact on autocorrelation decay and phase transitions, using a Markovian transformation.

Findings

Exponential decay kernels lead to stationary, mean-reverting processes.

Power-law decay kernels cause phase transitions and change autocorrelation decay behavior.

Power law exponents change discontinuously at the critical point.

Abstract

P\'{o}lya urn is a stochastic process in which balls are randomly drawn from an urn of red and blue balls, and balls of the same color as the drawn balls are added. The probability of a ball of a certain color being drawn is equal to the percentage of balls of that color in the urn. We introduce arbitrary memory kernels to modify this probability. If the memory kernel decays exponentially, it is a stationary process and is mean-reverting. If the memory kernel decays by a power-law, a phase transition occurs and the asymptotic behavior of the autocorrelation function changes. An auxiliary field variable is introduced to transform the process Markovian and the field obeys a multivariate Ornstein-Uhlenbeck process. The exponents of the power law are estimated for the decay of the leading and subleading terms of the autocorrelation function. It is shown that the power law exponents changes discontinuously at the critical point.