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Fluctuations of the empirical quantiles of independent Brownian motions

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  • Swanson, Jason

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.

Suggested Citation

  • Swanson, Jason, 2011. "Fluctuations of the empirical quantiles of independent Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 479-514, March.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:3:p:479-514
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    References listed on IDEAS

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    1. Tomasz Bojdecki & Luis G. Gorostiza & Anna Talarczyk, 2004. "Sub-fractional Brownian motion and its relation to occupation times," RePAd Working Paper Series lrsp-TRS376, Département des sciences administratives, UQO.
    2. Bojdecki, Tomasz & Gorostiza, Luis G. & Talarczyk, Anna, 2004. "Sub-fractional Brownian motion and its relation to occupation times," Statistics & Probability Letters, Elsevier, vol. 69(4), pages 405-419, October.
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    Cited by:

    1. James Kuelbs & Joel Zinn, 2015. "Empirical Quantile Central Limit Theorems for Some Self-Similar Processes," Journal of Theoretical Probability, Springer, vol. 28(1), pages 313-336, March.
    2. Daniel Harnett & Arturo Jaramillo & David Nualart, 2019. "Symmetric Stochastic Integrals with Respect to a Class of Self-similar Gaussian Processes," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1105-1144, September.

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