How large must be the difference between local time and mesure du voisinage of Brownian motion?
Let W(t), \(t\geq 0\), be a Wiener process on R and L(x,t) be the corresponding local time. The mesure du voisinage, M(x,t), of W is \(\lim_{h\downarrow 0}h^{1/2}N(h,x,t)\), where N(h,x,t) is the number of excursions of W away from x of length greater than h which are completed by time t. It is well known that \(M(x,t)=\sqrt{2/\pi}L(x,t)\) a.s. A result by two of the authors (Csörgö and Révész), unfortunately only available in a technical report, studies how quickly the limit defining M approaches \(\sqrt{2/\pi}L\). Specifically, for any fixed \(t>0\), \[ (1)\quad \lim_{h\downarrow 0}h^{-1/4}(-\log \quad h)^{-1}\sup_{(x,t)\in R\times [0,t']}| h^{1/2}N(h,x,t)-\sqrt{2/\pi}L(x,t)| =0,\quad a.s. \] In this paper the following result, related to how good an estimate (1) is for the rate of convergence, is shown. Theorem: For any \(\epsilon >0\) there exists \(\delta =\delta (\epsilon)>0\) and \(h_ 0=h_ 0(\epsilon)\) such that \[ P(h^{-1/4}| h^{1/2}N(h,0,1)-\sqrt{2/\pi}L(0,1)| >\delta)\geq 1-\epsilon \] if \(0<h<h_ 0\).
scientific article; zbMATH DE number 3971917
scientific article