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

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 a certain self-similar Gaussian process. In particular, the randomized occupancy process and its components converge jointly to a 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 a fractional Brownian motion with Hurst index $H\in (0,1/2)$. The decomposition in the latter case is a special case of the decomposition of bi-fractional Brownian motions recently investigated by Lei and Nualart (2009). The randomized odd-occupancy process can also be viewed as a correlated random walk, and in particular as a complement to the model recently introduced by Hammond and Sheffield (2013) as discrete analogues of fractional Brownian motions.