Philippe Berthet Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse Cedex 9, France

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Consider the set Θn of all an-sized increment processes of the uniform empirical process αn on [0, 1]. We assume that an ↓ 0, nan ↑ ∞, dn = nan(log n)−1 → ∞ and nan(log n)−7/3 = O(1). In Berthet (1996, 2005) the fourth assumption was shown to be critical with respect to the pointwise rates of convergence in the functional law of Deheuvels and Mason (1992) for Θn because strong approximation methods become ineffective at such a small scale an. We are now able to study directly these small empirical increments and compute the exact rate of clustering of Θn to any Strassen function having Lebesgue derivative of bounded variation by making use of a sharp small deviation estimate for a Poisson process of high intensity due to Shmileva (2003a). It turns out that the best rates are of order dn1/4(log n)−1 and are faster than in the Brownian case whereas the slowest rates are of order dn−1/2 and correspond to the apparently crude ones obtained in Berthet (2005) by means of Gaussian small ball probabilities. These different sharp properties of the empirical and Brownian paths imply an almost sure lower bound in the strong invariance principle and provide a new insight into the famous KMT approximation of αn.

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Periodica Mathematica Hungarica
Language English
Size B5
Year of
per Year
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Founder Bolyai János Matematikai Társulat - János Bolyai Mathematical Society
H-1055 Budapest, Hungary Falk Miksa u. 12.I/4.
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Chief Executive Officer, Akadémiai Kiadó
ISSN 0031-5303 (Print)
ISSN 1588-2829 (Online)

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