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On the coverage of strassen-type sets by sequences of Wiener processes

Authors:	Paul Deheuvels Pál Révész

Institution:	(1) L.S.T.A., Université Paris VI, 7 avenue du Château, 92340 Bourg-la-Reine, France;(2) Institut für Statistik und Wahrscheinlichkeitstheorie, T.U. Wien, Wiedner Hauptstrasse 6-10/107, A 1040 Wien, Austria

Abstract:	LetW ₁,W ₂,... be a sequence of Wiener processes and let K_T1 be a function ofT1. We consider the limiting behavior asT of the random set of functions defined by $\mathbb{F}_T = \{ (2T\{ \log K_T + \log \log T\} )^{ - 1/2} W_i (Ts),0 \leqslant s \leqslant 1,1 \leqslant i \leqslant k_T$ . Under suitable assumptions imposed uponK _T, we show that $\mathbb{F}_T$ covers asymptotically (in the sense of the Hausdorff set-metric induced by the sup-norm distance) Strassen-type sets equal, up to a multiplicative constant, to the limit set of functions obtained in the classical functional law of the iterated logarithm. Extensions of these results to arrays and increments of Wiener processes in the range studied by Book and Shore(²) are also provided.

Keywords:	Functional laws of the iterated logarithm Wiener processes
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