# On the Local Time of the Brownian Bridge

• Lajos Takács
Chapter
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 19)

## Abstract

Let {η(t), 0 ≤ t} ≤ 1} be a standard Brownian bridge. We have for 0 < t < 1, where
$$\Phi (x) = \frac{1}{{\sqrt {{2\pi }} }}\int_{{ - \infty }}^{x} {{{e}^{{ - u \frac{2}{2} }}}} du$$
(4.1)
is the normal distribution function. We define
$$\tau (\alpha ) = \mathop{{\lim }}\limits_{{\varepsilon \to 0}} \frac{1}{\varepsilon }{\text{measure\{ }}t:\alpha \leqslant \eta (t) < \alpha + \varepsilon ,0 \leqslant t \leqslant 1{\text{\} }}$$
(4.2)
for any real α. The limit (4.2) exists with probability one, and τ(α) is a nonnegative random variable that is called the local time at level α. We have
$$P\{ \tau (\alpha ) \leqslant x\} = 1 - {{e}^{{ - (2|\alpha | + x) \frac{2}{2} }}}$$
(4.3)
for x ≥ 0. The notion of local time was introduced by P. Lévy [9,10] in 1939 (see also [15] and [6]).

## Keywords

Random Walk Local Time Random Mapping Normal Distribution Function Empirical Distribution Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

1. [1]
Aldous, D., and Pitman, J. Brownian bridge asymptotics for random mappings. Random Structures Algorithms 5, 487–512, 1994.
2. [2]
Charlier, C. V. L. Application de la théorie des probabilités à l’astronomie. Gauthier-Villars, Paris, 1931.Google Scholar
3. [3]
Donsker, M. D. An invariance principle for certain probability limit theorems. Four Papers on Probability. Memoirs Am. Math. Soc. 6, 1–12, 1951.Google Scholar
4. [4]
Fréchet, M., and Shohat, J. A proof of the generalized second-limit theorem in the theory of probability. Trans. Am. Math. Soc. 33, 533–543, 1931.
5. [5]
Gikhman, I. I., and Skorokhod, A. V. Introduction to the Theory of Random Processes. W.B. Saunders, Philadelphia, 1969.Google Scholar
6. [6]
Itô, K., and McKean, H. P. Jr. Diffusion Processes and their Sample Paths. Springer-Verlag, Berlin, 1965.
7. [7]
Knight, F. B. Random walk and a sojourn density process of Brownian motion. Trans. Am. Math. Soc. 109, 56–86, 1965.
8. [8]
Kolmogoroff, A. Sulla determinazione empirica di una legge di distribuzione. Giorn. Ist. Ital. Attuari 4, 83–91, 1933.Google Scholar
9. [9]
Lévy, P. Sur certains processus stochastiques homogènes. Compositio Math. 7, 283–339, 1940.
10. [10]
Lévy, P. Processus Stochastiques et Mouvement Brownien, 2nd ed. Gauthier-Villars, Paris, 1965.
11. [11]
Proskurin, G. V. On the distribution of the number of vertices in strata of a random mapping. Theor. Prob. Appl. 18, 803–808, 1973.
12. [12]
Smirnov, N. V. On the estimation of the discrepancy between empirical curves of distribution for two independent samples. Bull. Math. l’Univ. Moscou, Série Internationale, 2(2), 3–16, 1939.Google Scholar
13. [13]
Smirnov, N. V. On deviations of the empirical distribution function. (In Russian.) Mat. Sbornik 6, 3–26, 1939.
14. [14]
Takács, L. Fluctuation problems for Bernoulli trials. SIAM Rev. 21, 222–228, 1979.
15. [15]
Trotter, H. A property of Brownian motion paths. Illinois J. Math. 2, 425–433, 1958.
16. [16]
Wolfram, S. Mathematica. A System for Doing Mathematics by Computer, 2nd ed. Addison-Wesley, Redwood City, CA, 1991.Google Scholar

© Springer Science+Business Media New York 1999

## Authors and Affiliations

• Lajos Takács

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