Fluctuations of the empirical quantiles of independent Brownian motions

Jason Swanson

Stochastic Processes and their Applications, 2011, vol. 121, issue 3, 479-514

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-->[alpha][set membership, variant](0,1). This sequence converges in probability to q(t), the [alpha]-quantile of the law of Bj(t). We first show convergence in law in C[0,[infinity]) of Fn=n1/2(Qn-q). We then investigate properties of the limit process F, including its local covariance structure, and Hölder-continuity and variations of its sample paths. In particular, we find that F has the same local properties as fBm with Hurst parameter H=1/4.

Keywords: Quantile; process; Order; statistics; Fluctuations; weak; convergence; Fractional; Brownian; motion; Quartic; variation (search for similar items in EconPapers)
Date: 2011
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Citations: View citations in EconPapers (2)

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