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Mesure du voisinage and occupation density. (English) Zbl 0581.60059

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.

MSC:

60J55 Local time and additive functionals
60J65 Brownian motion
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[1] Csörgo, M., Révész, P.: Strong approximations in probability and statistics. Budapest: Akadémiai Kiadó and New York: Academic Press 1981
[2] Csörgo, M., Révész, P.: On strong invariance for local time of partial sums. In: Tech. Rep. Ser. Lab. Res. Statist. Probab. 37, 25–72 (1984) Carleton University, and Stochastic Process. Appl. 20, 59–84 (1985)
[3] Itô, K., McKean, H.P.: Diffusion processes and their sample paths. Berlin-Heidelberg-New York: Springer 1965 · Zbl 0127.09503
[4] Knight, F.B.: Brownian local time and taboo processes. Trans. Am. Math. Soc. 143, 173–185 (1969) · Zbl 0187.41203 · doi:10.1090/S0002-9947-1969-0253424-7
[5] Knight, F.B.: Essentials of Brownian motion and diffusion. Math. Survey No. 18. Amer. Math. Soc. Providence, Rhode Island 1981 · Zbl 0458.60002
[6] Perkins, E.: A global intrinsic characterization of Brownian local time. Ann. Probab. 9, 800–817 (1981) · Zbl 0469.60081 · doi:10.1214/aop/1176994309
[7] Rényi, A.: Probability theory. Budapest: Akadémiai Kiadó 1970
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