From infinite urn schemes to self-similar stable processes
Durieu, Olivier; Samorodnitsky, Gennady; Wang, Yizao
We investigate the randomized Karlin model with parameter beta in (0,1), which is based on an infinite urn scheme. It has been shown before that when the randomization is bounded, the so-called odd-occupancy process scales to a fractional Brownian motion with Hurst index beta/2 in (0,1/2). We show here that when the randomization is heavy-tailed with index alpha in (0,2), then the odd-occupancy process scales to a (beta/alpha)-self-similar symmetric alpha-stable process with stationary increments.
The first author would like to thank the hospitality and financial support from Taft Research Center and Department of Mathematical Sciences at University of Cincinnati, for his visits in 2016 and 2017. The second author's research was partially supported by NSF grant DMS-1506783 and the ARO grant W911NF-12-10385 at Cornell University. The third author's research was partially supported by the NSA grants H98230-14-1-0318 and H98230-16-1-0322, the ARO grant W911NF-17-1-0006, and Charles Phelps Taft Research Center at University of Cincinnati.
inifinite urn scheme; regular variation; stable process; self-similar process; functional central limit theorem