Electronic Journal of Probability

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

Olivier Durieu and Yizao Wang

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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

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

Received: 19 August 2015
Accepted: 4 July 2016
First available in Project Euclid: 26 July 2016

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Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60G22: Fractional processes, including fractional Brownian motion
Secondary: 60G15: Gaussian processes 60G18: Self-similar processes

infinite urn scheme regular variation functional central limit theorem self-similar process fractional Brownian motion bi-fractional Brownian motion decomposition symmetrization

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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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