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Ergodic properties of Brownian Motion

Published online by Cambridge University Press:  20 January 2009

Olabisi O. Ugbebor
Olabisi O. Ugbebor
Affiliation:
University of IbadanIbadanNigeria
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Since Brownian motion is point recurrent in R1, recurrent in R2 and transient in Rn, n ≧ 3 (see (7)), it follows that the total time spent in a bounded open set in R1 or R2 is unbounded. With the following ergodic theorems for Brownian motion in R1 and R2 as motivation, we examine the rate of convergence in these theorems. Note that there is no ergodic property in Rn for n ≧ 3 since Brownian motion is not dense there.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 23 , Issue 3 , October 1980 , pp. 331 - 340
Copyright
Copyright © Edinburgh Mathematical Society 1980

References

REFERENCES

(1)Apostol, T. M., Mathematical Analysis (Addison-Wesley Publ. Co. Inc, 1957).Google Scholar
(2)Ciesielski, Z. and Taylor, S. J., First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path, Trans. Amer. Math. Soc. 103 (1962), 434–450.CrossRefGoogle Scholar
(3)Derman, C., Ergodic Property of the Brownian motion process, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 1155–1158.CrossRefGoogle ScholarPubMed
(4)Doob, J. L., Stochastic Processes (John Wiley and Sons Inc, 1953).Google Scholar
(5)Ibragimov, I. A., Some limit theorems for stationary processes, Theory of Prob. and its Applns. 7 (1962), 349–382.CrossRefGoogle Scholar
(6)Kingman, J. F. C. and Taylor, S. J., Introduction to Measure and Probability (Cambridge University Press, 1966).CrossRefGoogle Scholar
(7)Levy, P., Processus Stochastiques et Mouvement Brownien (Gauthier Villars, Paris, 1948).Google Scholar
(8)Maruyama, G. and Tanaka, H., Ergodic property of N-dimensional recurrent Markov processes, Memoirs Fac. Sci. Kyushu Univ. Ser. A 13 (1959), No. 2.Google Scholar
(9)Port, S. and Stone, C., Classical Potential theory and Brownian motion, Logarithmic potentials and planar Brownian motion, Proc. 6th Berkeley Symp. Math. Statist. Prob. 3 (1972), 143–176, 177–192.Google Scholar
(10)Reznik, M. Kh., The law of the iterated logarithm for some classes of stationary processes, Teor. Veroj. Primenen 13 (1968), 642–656.Google Scholar