Title

Fluctuations Of The Empirical Quantiles Of Independent Brownian Motions

Keywords

Fluctuations weak convergence; Fractional Brownian motion; Order statistics; Quantile process; Quartic variation

Abstract

We consider iid Brownian motions, Bj(t), where Bj(0) has a rapidly decreasing, smooth density function f. The empirical quantiles, or pointwise order statistics, are denoted by Bj:n(t), and we consider a sequence Qn(t)=Bj(n):n(t), where j(n)n→α∈(0,1). This sequence converges in probability to q(t), the α-quantile of the law of Bj(t). We first show convergence in law in C[0,∞) of Fn=n12(Qn-q). We then investigate properties of the limit process F, including its local covariance structure, and Hlder-continuity and variations of its sample paths. In particular, we find that F has the same local properties as fBm with Hurst parameter H=14. © 2010 Elsevier B.V. All rights reserved.

Publication Date

3-1-2011

Publication Title

Stochastic Processes and their Applications

Volume

121

Issue

3

Number of Pages

479-514

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.spa.2010.11.012

Socpus ID

78751591174 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/78751591174

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