Summary
Many results are known about the convergence of some processes to Brownian local time. Among such processes are the process of “occupation times” of Brownian motion, the number of downcrossings of Brownian motion over smaller and smaller intervals before timet, the number of visits of the recurrent integer-valued random walk to some point duringn steps and others. In this paper we consider the asymptotic behaviour of the differences between Brownian local time and some of the processes which converge to it.
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Barlow, M.T.: Continuity of local times for Lévy processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.69, 243–250 (1985)
Billingsley, P.: Convergence of probability measures. New York: Wiley 1968
Borodin, A.N.: On distribution of integral type functionals of Brownian motion. Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V.A. Steklova AN SSSR119, 19–38 (1982)
Borodin, A.N.: Limit theorems for local times. In: Proceedings of XYI All-Union school on probability theory and math. statistics, pp. 114–115. Tbilisi: Metsniereba 1982
Borodin, A.N.: On the character of convergence to Brownian local time. Dokl. USSR Academy of Sciences269, 784–788 (1983)
Borodin, A.N.: On the distribution of the supremum of increments of Brownian local time. Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V.A. Steklova AN SSSR142, 6–24 (1985)
Chacon, R.V., LeJan, Y., Perkins, E., Taylor, S.J.: Generalised arclength for Brownian motion and Lévy processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.57, 197–212 (1981)
Doob, J.L.: Stochastic Processes. New York: Wiley 1953
Getoor, R.K.: Another limit theorem for local time. Z. Wahrscheinlichkeitstheor. Verw. Geb.34, 1–10 (1976)
Ito, K., McKean, H.P.: Diffusion processes and their sample paths. Berlin-Heidelberg-New York: Springer 1965
Kac, M.: On distribution of certain Wiener functionals. Trans. Am. Math. Soc.65, 1–13 (1949)
Kasahara, Y.: On Lévy's downcrossing theorem' Proc. Japan Acad. Ser. A, Math. Sci.56, N10, 455–458 (1980)
Lévy, P.: Construction du processus de W. Feller et H.P. McKean en partant du mouvement Brownian. In: Probability and Statistics (the Harald Cramér volume), pp. 162–174. Stockholm: Almqvist and Wiksell 1959
Ray, D.B.: Sojourn times of a diffusion process. Ill. J. Math.7, 615–630 (1963)
Skorokhod, A.V., Slobodenyuk, N.P.: Limit theorems for random walks. Kiev: Naukova Dumka 1970
Trotter, H.F.: A property of Brownian motion paths. Ill. J. Math.2, 425–433 (1958)
Wang, A.T.: Generalized Ito's formula and additive functionals of Brownian motion. Z. Wahrscheinlichkeitstheor. Verw. Geb.41, 153–159 (1977)
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Borodin, A.N. On the character of convergence to Brownian local time. I. Probab. Th. Rel. Fields 72, 231–250 (1986). https://doi.org/10.1007/BF00699105
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DOI: https://doi.org/10.1007/BF00699105