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Functional laws of the iterated logarithm for large increments of empirical and quantile processes

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  • Deheuvels, Paul

Abstract

Let {[alpha]n(t),0[less-than-or-equals, slant]t[less-than-or-equals, slant]1} and {[beta]n(t),0[less-than-or-equals, slant]t[less-than-or-equals, slant]1} be the empirical and quantile processes generated by the first n observations from an i.i.d. sequence of random variables with a uniform distribution on (0, 1). Let 0 0 and (log(1/hn))/log log n --> c [epsilon][0,[infinity]) as n-->[infinity]. Under suitable additional regularity conditions imposed upon hn, we prove functional laws of the iterated logarithm for . We present applications of these results to nonparametric densityestimation, and prove a conjecture of Shorack and Wellner (1986) concerning the limiting behaviour of the maximal increments of [alpha]n and [beta]n.

Suggested Citation

  • Deheuvels, Paul, 1992. "Functional laws of the iterated logarithm for large increments of empirical and quantile processes," Stochastic Processes and their Applications, Elsevier, vol. 43(1), pages 133-163, November.
    • Handle: RePEc:eee:spapps:v:43:y:1992:i:1:p:133-163
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    Citations

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    Cited by:

    1. Philippe Berthet, 1997. "On the Rate of Clustering to the Strassen Set for Increments of the Uniform Empirical Process," Journal of Theoretical Probability, Springer, vol. 10(3), pages 557-579, July.
    2. Labrador, Boris, 2008. "Strong pointwise consistency of the kT -occupation time density estimator," Statistics & Probability Letters, Elsevier, vol. 78(9), pages 1128-1137, July.
    3. Uwe Einmahl & David M. Mason, 2000. "An Empirical Process Approach to the Uniform Consistency of Kernel-Type Function Estimators," Journal of Theoretical Probability, Springer, vol. 13(1), pages 1-37, January.
    4. Djamal Louani & Alain Lucas, 2003. "Fractal Dimensions for Some Increments of the Uniform Empirical Process," Journal of Theoretical Probability, Springer, vol. 16(1), pages 59-86, January.
    5. Paul Deheuvels & Sarah Ouadah, 2013. "Uniform-in-Bandwidth Functional Limit Laws," Journal of Theoretical Probability, Springer, vol. 26(3), pages 697-721, September.
    6. Varron, Davit, 2011. "Some new almost sure results on the functional increments of the uniform empirical process," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 337-356, February.
    7. Berthet, Philippe, 2005. "Inner rates of coverage of Strassen type sets by increments of the uniform empirical and quantile processes," Stochastic Processes and their Applications, Elsevier, vol. 115(3), pages 493-537, March.
    8. Dindar, Zacharie, 2000. "Random fractals generated by oscillations of the uniform empirical process," Statistics & Probability Letters, Elsevier, vol. 50(3), pages 285-291, November.
    9. Dindar, Zacharie, 2003. "Some more results on increments of the partially observed empirical process," Statistics & Probability Letters, Elsevier, vol. 64(1), pages 25-37, August.
    10. Paul Deheuvels, 1998. "On the Approximation of Quantile Processes by Kiefer Processes," Journal of Theoretical Probability, Springer, vol. 11(4), pages 997-1018, October.
    11. Varron, Davit, 2008. "Some asymptotic results on density estimators by wavelet projections," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2517-2521, October.
    12. Einmahl, J.H.J. & Deheuvels, P., 2000. "Functional limit laws for the increments of Kaplan-Meier product-limit processes and applications," Other publications TiSEM ac9bbdc0-62f8-4b48-9a84-1, Tilburg University, School of Economics and Management.

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