Mesure du voisinage and occupation density
Let W(t) be a standard Wiener process with occupation density (local time) \(\eta\) (x,t). P. Lévy showed that for each x, \(\eta\) (x,t) is a.s. equal to the ''mesure du voisinage'' of W, i.e., to the limit as h approaches zero of \(h^{1/2}\) times N(h,x,t), the number of excursions from x, exceeding h in length, that are completed by W up to time t. Recently E. Perkins showed that the exceptional null sets, which may depend on x, can be combined into a single null set off which the above convergence is uniform in x. The main aim of the present paper is to estimate the rate of convergence in Perkins' theorem as h goes to zero. We also investigate the connection between N and \(\eta\) in the case when we observe a Wiener process through a long time t and consider the number of long (but much shorter than t) excursions.
scientific article; zbMATH DE number 3930067