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Inner rates of coverage of Strassen type sets by increments of the uniform empirical and quantile processes

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  • Berthet, Philippe

Abstract

We establish Chung-Mogulskii type functional laws of the iterated logarithm for medium and large increments of the uniform empirical and quantile processes. This gives the ultimate sup-norm distance between various sets of properly normalized empirical increment processes and a fixed function of the relevant cluster sets. Interestingly, we obtain the exact rates and constants even for most functions of the critical border of Strassen type balls and further introduce minimal entropy conditions on the locations of the increments under which the fastest rates are achieved with probability one. Similar results are derived for the Brownian motion and other related processes.

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  • 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.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:3:p:493-537
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    References listed on IDEAS

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    1. Mason, David M., 1984. "A strong limit theorem for the oscillation modulus of the uniform empirical quantile process," Stochastic Processes and their Applications, Elsevier, vol. 17(1), pages 127-136, May.
    2. Shi, Z., 1991. "A generalization of the Chung-Mogul'skii law of the iterated logarithm," Journal of Multivariate Analysis, Elsevier, vol. 37(2), pages 269-278, May.
    3. 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.
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    Cited by:

    1. 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.

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