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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
##### Keywords:
Brownian motion; rates of clustering
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##### References:
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