From infinite urn schemes to self-similar stable processes

TL;DR

This paper extends the understanding of infinite urn schemes by demonstrating that with heavy-tailed randomization, the odd-occupancy process converges to a self-similar symmetric alpha-stable process, broadening the class of limit processes.

Contribution

It introduces a new scaling limit for the odd-occupancy process under heavy-tailed randomization, showing convergence to a symmetric alpha-stable process with self-similarity.

Findings

Heavy-tailed randomization leads to alpha-stable process limits.

The odd-occupancy process exhibits self-similarity with index beta/alpha.

The model generalizes previous fractional Brownian motion results.

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.