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Rates of clustering in Strassen’s LIL for Brownian motion. (English) Zbl 0724.60034
For standard Brownian motion \(\{W(t): 0\leq t<\infty \}\) let \(\eta_ n(t)=W(nt)(2nL_ 2n)^{-1/2}\), where \(t\in [0,1]\), \(n\in {\mathbb{N}}\) and \(L_ mn=L(L_{m-1}n)\), \(Lx=\max \{1,\log_ ex\}\). For a set \(A\subset C[0,1]\) denote \(A^{\epsilon}=\{g\in C[0,1]:\;\inf_{f\in A}\| g- f\| <\epsilon \},\) where \(\epsilon >0\) and \(\| \cdot \|\) is the uniform norm. The following generalizations of K. Grill’s result [Probab. Theory Relat. Fields 74, 583-589 (1987; Zbl 0592.60022)] are obtained. If \({\mathcal K}\) is the Strassen’s set, \(\epsilon_ n=\gamma (L_ 3n/L_ 2n)^{2/3}\), then for \(\gamma >0\) sufficiently large, \(P(\eta_ n\in {\mathcal K}^{\epsilon_ n}\) eventually)=1. If \(\epsilon_ n=\gamma (L_ 2n)^{-2/3}\), then for \(\gamma >0\) sufficiently small \(P(\eta_ n\not\in {\mathcal K}^{\epsilon_ n}\quad i.o.)=1.\) Introduce the random sets \(E_{\xi (n),n}=\{W(k(\cdot))(2kL_ 2n)^{-1/2}:\;\xi (n)\leq k\leq n\}.\) If \(\xi (n)\leq n^{1/4}\) and \(\epsilon_ n=\gamma (L_ 2n)^{-2/3}\), then for \(\gamma >0\) sufficiently large, \(P({\mathcal K}\subseteq (E_{\xi (n),n})^{\epsilon_ n}\) eventually)=1 and for \(\gamma >0\) sufficiently small \(P({\mathcal K}\subseteq (E_{1,n})^{\epsilon_ n}\) eventually)=0. Assume \(L_ 2\xi (n)\sim L_ 2n\) as \(n\to \infty\). Then for \(\epsilon_ n=\gamma (L_ 3n/L_ 2n)^{2/3}\) with \(\gamma >0\) sufficiently large, \(P(E_{\xi (n),n}\subseteq {\mathcal K}^{\epsilon_ n}\) eventually)=1.

MSC:
60F15 Strong limit theorems
60J65 Brownian motion
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