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Perman, Mihael; Wellner, Jon A.

On the distribution of Brownian areas. (English) Zbl 0870.60035

Ann. Appl. Probab. 6, No. 4, 1091-1111 (1996).
Summary: We study the distributions of the areas under the positive parts of a Brownian motion process \(B\) and a Brownian bridge process \(U\): with \(A^+=\int^1_0B^+(t)dt\) and \(A^+_0= \int^1_0U^+(t)dt\), we use excursion theory to show that the Laplace transforms \(\Psi^+(s)= E\exp(-sA^+)\) and \(\Psi^+_0(s)= E\exp(-sA^+_0)\) of \(A^+\) and \(A^+_0\) satisfy \[ \int^\infty_0 e^{-\lambda s}\Psi^+(\sqrt 2s^{3/2})ds= {\lambda^{-1/2}Ai(\lambda)+ (1/3-\int^\lambda_0A_i(t)dt)\over \sqrt\lambda Ai(\lambda)-Ai'(\lambda)} \] and \[ \int^\infty_0 {e^{-\lambda s}\over\sqrt s} \Psi^+_0(\sqrt 2 s^{3/2})ds= 2\sqrt\pi {Ai(\lambda)\over\sqrt\lambda Ai(\lambda)- Ai'(\lambda)}, \] where \(Ai\) is Airy’s function. At the same time, our approach via excursion theory unifies previous calculations of this type due to Kac, Groeneboom, Louchard, Shepp and Takács for other Brownian areas. Similarly, we use excursion theory to obtain recursion formulas for the moments of the “positive part” areas. We have not yet succeeded in inverting the double Laplace transforms because of the structure of the function appearing in the denominators, namely, \(\sqrt\lambda Ai(\lambda)- Ai'(\lambda)\).

Cited in 1 Review
Cited in 21 Documents

MSC:

60G15 Gaussian processes
60G99 Stochastic processes
60E05 Probability distributions: general theory

Keywords:

absolute value; Airy functions; area; asymptotic distribution; Brownian bridge; Brownian excursion; Brownian motion; Feynman-Kac; inversion; moments; positive part; recursion formulas
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References:

[1] ABRAMOWITZ, M. and STEGUN, I. A. 1965. Handbook of Mathematical Functions. Dover, New York. Z. · Zbl 0515.33001
[2] BARLOW, M., PITMAN, J. M. and YOR, M. 1989. Une extension multidimensionnelle de la loi de l’arc sinus. Seminaire de Probabilites XXIII. Lecture Notes in Math. 1372 294 314. \' Śpringer, New York. Z. · Zbl 0738.60072
[3] BIANE, PH. and YOR, M. 1987. Valuers principales associees aux temps locaux Browniens. Bull. Ści. Math. 111 23 101. Z. · Zbl 0619.60072
[4] BIANE, PH. and YOR, M. 1988. Sur la loi des temps locaux browniens pris en un temps exponentiel. Seminaire de Probabilites XXII. Lecture Notes in Math. 1321 454 465. \' Śpringer, New York. Z. · Zbl 0652.60081
[5] BIRNBAUM, Z. W. and TANG, V. K. T. 1964. Two simple distribution-free tests of goodness of fit. Rev. Internat. Statist. Inst. 32 2 13. Z. · doi:10.2307/1401970
[6] BORODIN, A. N. 1984. Distribution of integral functionals of a Brownian motion. J. Soviet Math. 27 3005 3022. Z. · Zbl 0549.60074 · doi:10.1007/BF01843545
[7] CHAPMAN, D. G. 1958. A comparative study of several one-sided goodness-of-fit tests. Ann. Math. Statist. 29 655. Z. · Zbl 0086.12702 · doi:10.1214/aoms/1177706529
[8] CIFARELLI, D. M. 1975. Contributi intorno ad un test per l’omogeneita tra du campioni. G. Z. Econom. Ann. Econ. N.S. 34 233 249. Z.
[9] DARLING, D. A. 1983. On the supremum of a certain Gaussian process. Ann. Probab. 11 803 806. Z. · Zbl 0515.60044 · doi:10.1214/aop/1176993527
[10] DURRETT, R. T. and IGLEHART, D. L. 1977. Functionals of Brownian meander and excursion. Ann. Probab. 5 130 135. Z. · Zbl 0356.60035 · doi:10.1214/aop/1176995896
[11] Dy NKIN, E. B. 1961. Some limit theorems for sums of independent random variables with infinite expectations. Selected Translations in Mathematical Statistics and Probability, IMS AMS 1 171 189. Z. · Zbl 0112.10105
[12] FELLER, W. 1971. An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York. Z. · Zbl 0219.60003
[13] GETOOR, R. K. 1979. Excursions of a Markov process. Ann. Probab. 7 244 266. Z. · Zbl 0399.60069 · doi:10.1214/aop/1176995086
[14] GROENEBOOM, P. 1985. Estimating a monotone density. Proceedings of the Berkeley Conference Z. in Honor of Jerzy Ney man and Jack Kiefer L. M. Le Cam and R. A. Olshen, eds. 2. Univ. California Press, Berkeley. Z. · Zbl 1373.62144
[15] GROENEBOOM, P. 1989. Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 79 109. Z. · doi:10.1007/BF00343738
[16] ITO, K. and MCKEAN, H. P. 1974. Diffusion Processes and Their Sample Paths, 2nd ed. Springer, Berlin. Z. · Zbl 0285.60063
[17] JOHNSON, B. MCK. and KILLEEN, T. 1983. An explicit for the c.d.f. of the L norm of the 1 Brownian bridge. Ann. Probab. 11 807 808. Z. · Zbl 0516.60044 · doi:10.1214/aop/1176993528
[18] KAC, M. 1946. On the average of a certain Wiener functional and a related limit theorem in calculus of probability. Trans. Amer. Math. Soc. 59 401 414. Z. JSTOR: · Zbl 0063.03092 · doi:10.2307/1990263
[19] KAC, M. 1951. On some connections between probability theory and differential and integral equations. Proc. Second Berkeley Sy mp. Math. Statist. Probab. 1 189 215. Univ. California Press, Berkeley. Z. · Zbl 0045.07002
[20] LEVY, P. 1948. Processus Stochastiques et mouvement Brownien. Gauthiers, Paris. \' · Zbl 0034.22603
[21] LOUCHARD, G. 1984a. Kac’s formula, Levy’s local time and Brownian excursion. J. Appl. Ṕrobab. 21 479 499. Z. · Zbl 0551.60086 · doi:10.2307/3213611
[22] LOUCHARD, G. 1984b. The Brownian excursion area. Comput. Math. Appl. 10 413 417. ErraZ. tum: A12 1986 375. Z. · Zbl 0569.65097
[23] NASAROW, N. 1931. Ueber die Entwicklung einer beliebigen Funktion nach Laguerreschen Poly nomen. Math. Z. 33 481 487. Z. · Zbl 0001.39601 · doi:10.1007/BF01174366
[24] REVUZ, D. and YOR, M. 1994. Continuous Martingales and Brownian Motion, 2nd ed. Springer, New York. Z. · Zbl 0804.60001
[25] RICE, S. O. 1982. The integral of the absolute value of the pinned Wiener process. Ann. Probab. 10 240 243. Z. · Zbl 0479.60080 · doi:10.1214/aop/1176993927
[26] RIEDWy L, H. 1967. Goodness of fit. J. Amer. Statist. Assoc. 62 390 398. Z. JSTOR: · doi:10.2307/2283969
[27] ROGERS, L. C. G. and WILLIAMS, D. 1987. Diffusions, Markov Processes and Martingales. Ito Calculus 2. Wiley, New York. Z. · Zbl 0977.60005
[28] SANSONE, G. 1959. Orthogonal Functions. Interscience, New York. Z. · Zbl 0084.06106
[29] SHEPP, L. A. 1982. On the integral of the absolute value of the pinned Wiener process. Ann. · Zbl 0479.60079 · doi:10.1214/aop/1176993926
[30] TAKACS, L. 1992a. Random walk processes and their various applications. In Probability Ź Theory and Applications. Essay s to the Memory of Jozsef Mogy orodi J. Galambos and \' \'. I. Katai, eds. 1 32. Kluwer, Dordrecht. Ź. · Zbl 0773.60068
[31] TAKACS, L. 1992b. Random walk processes and their application in order statistics. Ann. Appl. Ṕrobab. 2 435 459. Z. · Zbl 0754.62033 · doi:10.1214/aoap/1177005710
[32] TAKACS, L. 1993a. On the distribution of the integral of the absolute value of the Brownian ḿotion. Ann. Appl. Probab. 3 186 197. Z. · Zbl 0776.60051 · doi:10.1214/aoap/1177005514
[33] TAKACS, L. 1993b. Personal communication with J. A. Wellner. Ź.
[34] USPENSKY, J. V. 1927. On the development of arbitrary functions in series of Hermite’s and Laguerre’s poly nomials. Ann. Math. 28 593 619. Z. · JFM 53.0270.01
[35] WOLFRAM, S. 1991. Mathematica, a Sy stem for Doing Mathematics by Computer, 2nd ed. Addison-Wesley, Redwood City, CA.
[36] INSTITUTE FOR MATHEMATICS, physiCS UNIVERSITY OF WASHINGTON AND MECHANICS DEPARTMENT OF STATISTICS DEPARTMENT OF MATHEMATICS BOX 354322
[37] UNIVERSITY OF LJUBLJANA SEATTLE, WASHINGTON 98195-4322 JADRANSKA 19 E-MAIL: jaw@stat.washington.edu 61111 LJUBLJANA SLOVENIA E-MAIL: Mihael.Perman@fmf.uni-lj.si
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