Pólya urn with memory kernel and asymptotic behaviours of autocorrelation function

Abstract

Pólya urn is a successive random drawing process of a ball from an urn that contains red and blue balls, in which the ball is returned to the urn with an additional ball of the same colour. The probability to draw a red ball is equal to the ratio of the red balls in the urn. We introduce arbitrary memory kernels to modify the probability. When the memory kernel decays by a power-law, there occurs a phase transition in the asymptotic behavior of the autocorrelation function. We introduce an auxiliary field variable that obeys a stochastic differential equation with a common noise. We estimate the power-law exponents of the leading and sub-leading terms of the autocorrelation function. We show that the power-law exponents change discontinuously at the critical point.

From the multi-terms urn model to the self-exciting negative binomial distribution and Hawkes processes

“…Models describing these phenomena have correlations [4][5][6][7][8][9][10], and show several types of phase transitions. In our previous work, we investigated voting models for an information cascade [11][12][13][14][15][16][17][18]. This model is a type of urn process that represents the correlations and has two types of phase transitions.…”

“…In our previous study, we discussed the parameter estimation of the urn process, which has a correlation in the same term, and considered a multi-year case with a temporal correlation [17,18]. In this work, we introduce a similar multi-term urn process and discuss the relationship between the new urn process and the Hawkes process.…”