From infinite urn schemes to decompositions of self-similar Gaussian processes

TL;DR

This paper explores the connection between infinite urn schemes and self-similar Gaussian processes, establishing limit theorems that decompose these processes into fundamental components, with implications for models of correlated random walks.

Contribution

It introduces a natural randomization of occupancy processes and proves their convergence to decompositions of self-similar Gaussian processes, extending previous models and theories.

Findings

Randomized occupancy processes converge to decompositions of time-changed Brownian motion.

Randomized odd-occupancy processes converge to decompositions of fractional Brownian motion.

The decompositions relate to models of correlated random walks and fractional Brownian motion analogues.

Abstract

We investigate a special case of infinite urn schemes first considered by Karlin (1967), especially its occupancy and odd-occupancy processes. We first propose a natural randomization of these two processes and their decompositions. We then establish functional central limit theorems, showing that each randomized process and its components converge jointly to a decomposition of certain self-similar Gaussian process. In particular, the randomized occupancy process and its components converge jointly to the decomposition of a time-changed Brownian motion $\mathbb B(t^\alpha), \alpha\in(0,1)$, and the randomized odd-occupancy process and its components converge jointly to a decomposition of fractional Brownian motion with Hurst index $H\in(0,1/2)$. The decomposition in the latter case is a special case of the decompositions of bi-fractional Brownian motions recently investigated by Lei and Nualart (2009). The randomized odd-occupancy process can also be viewed as correlated random walks, and in particular as a complement to the model recently introduced by Hammond and Sheffield (2013) as discrete analogues of fractional Brownian motions.