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From infinite urn schemes to self-similar stable processes
Abstract
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.
Sponsorship
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.
Date Issued
2017Subject
inifinite urn scheme; regular variation; stable process; self-similar process; functional central limit theorem
Type
preprint