## Electronic Journal of Probability

### 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.

#### Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 43, 23 pp.

Dates
Accepted: 4 July 2016
First available in Project Euclid: 26 July 2016

https://projecteuclid.org/euclid.ejp/1469557136

Digital Object Identifier
doi:10.1214/16-EJP4492

Mathematical Reviews number (MathSciNet)
MR3530320

Zentralblatt MATH identifier
1346.60039

#### Citation

Durieu, Olivier; Wang, Yizao. From infinite urn schemes to decompositions of self-similar Gaussian processes. Electron. J. Probab. 21 (2016), paper no. 43, 23 pp. doi:10.1214/16-EJP4492. https://projecteuclid.org/euclid.ejp/1469557136

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