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On the Rate of Clustering to the Strassen Set for Increments of the Uniform Empirical Process

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  • Philippe Berthet

Abstract

We obtain outer rates of clustering in the functional laws of the iterated logarithm of Deheuvels and Mason(11) and Deheuvels,(7) which describe local oscillations of empirical processes. Considering increment sizes a n ↓ 0 such that na n ↑ ∞ and na n(log n)−7/3 → ∞ we show that the sets of properly rescaled increment functions cluster with probability one to the ε n-enlarged Strassen ball in B(0, 1) endowed with the uniform topology, where ε n ↓ 0 may be chosen so small as ε(log (1/a n) + log log n)−2/3 for any sufficiently large ε. This speed of coverage is reduced for smaller a n.

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  • 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.
    • Handle: RePEc:spr:jotpro:v:10:y:1997:i:3:d:10.1023_a:1022632825288
    DOI: 10.1023/A:1022632825288
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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. 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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