On the global asymptotic behavior of Brownian local time on the circle

Author:
E. Bolthausen

Journal:
Trans. Amer. Math. Soc. **253** (1979), 317-328

MSC:
Primary 60F05; Secondary 60J55

MathSciNet review:
536950

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Abstract: The asymptotic behavior of the local time of Brownian motion on the circle is investigated. For fixed time point *t* this is a (random) continuous function on . It is shown that after appropriate norming the distribution of this random element in converges weakly as . The limit is identified as where *B* is the Brownian bridge. The result is applied to obtain the asymptotic distribution of a Cramer-von Mises type statistic for the global deviation of the local time from the constant *t* on .

**[1]**J. R. Baxter and G. A. Brosamler,*Energy and the law of the iterated logarithm*, Math. Scand.**38**(1976), no. 1, 115–136. MR**0426178****[2]**Patrick Billingsley,*Convergence of probability measures*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0233396****[3]**R. M. Blumenthal and R. K. Getoor,*Markov processes and potential theory*, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968. MR**0264757****[4]**G. A. Brosamler,*A probabilistic solution of the Neumann problem*, Math. Scand.**38**(1976), no. 1, 137–147. MR**0408009****[5]**M. D. Donsker and S. R. S. Varadhan,*Asymptotic evaluation of certain Markov process expectations for large time. I. II*, Comm. Pure Appl. Math.**28**(1975), 1–47; ibid. 28 (1975), 279–301. MR**0386024****[6]**Avner Friedman,*Stochastic differential equations and applications. Vol. 1*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Probability and Mathematical Statistics, Vol. 28. MR**0494490****[7]**John G. Kemeny, J. Laurie Snell, and Anthony W. Knapp,*Denumerable Markov chains*, 2nd ed., Springer-Verlag, New York-Heidelberg-Berlin, 1976. With a chapter on Markov random fields, by David Griffeath; Graduate Texts in Mathematics, No. 40. MR**0407981****[8]**H. P. McKean,*Brownian local times*, Topics in Probability Theory, D. W. Stroock and S. R. S. Vardahan (editors), Courant Inst. Math. Sci., New York Univ., New York, 1973, pp. 59-92.**[9]**H. P. McKean Jr.,*Stochastic integrals*, Probability and Mathematical Statistics, No. 5, Academic Press, New York-London, 1969. MR**0247684****[10]**A. Rényi,*On the central limit theorem for the sum of a random number of independent random variables*, Acta Math. Acad. Sci. Hungar.**11**(1960), 97–102 (unbound insert) (English, with Russian summary). MR**0115204****[11]**Hiroshi Tanaka,*Certain limit theorems concerning one-dimensional diffusion processes.*, Mem. Fac. Sci. Kyusyu Univ. Ser. A. Math.**12**(1958), 1–11. MR**0097129**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1979-0536950-9

Keywords:
Brownian motion on the circle,
local time,
weak convergence

Article copyright:
© Copyright 1979
American Mathematical Society