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2016 From infinite urn schemes to decompositions of self-similar Gaussian processes
Olivier Durieu, Yizao Wang
Electron. J. Probab. 21: 1-23 (2016). DOI: 10.1214/16-EJP4492

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.

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Olivier Durieu. Yizao Wang. "From infinite urn schemes to decompositions of self-similar Gaussian processes." Electron. J. Probab. 21 1 - 23, 2016. https://doi.org/10.1214/16-EJP4492

Information

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

zbMATH: 1346.60039
MathSciNet: MR3530320
Digital Object Identifier: 10.1214/16-EJP4492

Subjects:
Primary: 60F17 , 60G22
Secondary: 60G15 , 60G18

Keywords: bi-fractional Brownian motion , Decomposition , fractional Brownian motion , functional central limit theorem , infinite urn scheme , regular variation , self-similar process , symmetrization

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