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On the rate of clustering to the Strassen set for increments of the uniform empirical process. (English) Zbl 0884.60029

Summary: We obtain outer rates of clustering in the functional laws of the iterated logarithm of P. Deheuvels and D. M. Mason [Ann. Probab. 20, No. 3, 1248-1287 (1992; Zbl 0760.60028)] and P. Deheuvels [Stochastic Processes Appl. 43, No. 1, 133-163 (1992; Zbl 0767.60028)] which describe local oscillations of empirical processes. Considering increment sizes \(a_n\downarrow 0\) such that \(na_n\uparrow\infty\) and \(na_n(\log n)^{-7/3}\to \infty\), we show that the sets of properly rescaled increment functions cluster with probability one to the \(\varepsilon_n\)-enlarged Strassen ball in \(B(0, 1)\) endowed with the uniform topology, where \(\varepsilon_n\downarrow 0\) may be chosen so small as \(\varepsilon(\log(1/a_n)+ \log\log n)^{-2/3}\) for any sufficiently large \(\varepsilon\). This speed of coverage is reduced for smaller \(a_n\).

MSC:

60F17 Functional limit theorems; invariance principles
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