An excursion approach to maxima of the Brownian bridge. (English) Zbl 1306.60115

Summary: Distributions of functionals of the Brownian bridge arise as limiting distributions in non-parametric statistics. In this paper we will give a derivation of distributions of extrema of the Brownian bridge based on excursion theory for Brownian motion. The idea of rescaling and conditioning on the local time has been used widely in the literature. In this paper it is used to give a unified derivation of a number of known distributions, and a few new ones. Particular cases of calculations include the distribution of the Kolmogorov-Smirnov statistic and the Kuiper statistic.


60J65 Brownian motion
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60J55 Local time and additive functionals
62E20 Asymptotic distribution theory in statistics
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[1] Barlow, M.; Pitman, J.; Yor, M., Une extension multidimensionnelle de la loi de l’arc sinus, (Séminaire de Probabilités, XXIII, Lecture Notes in Math., vol. 1372, (1989), Springer Berlin), 294-314 · Zbl 0738.60072
[2] Bertoin, J., (Lévy Processes, Cambridge Tracts in Mathematics, vol. 121, (1996), Cambridge University Press Cambridge)
[3] Borodin, A. N.; Salminen, P., (Handbook of Brownian Motion—Facts and Formulae, Probability and its Applications, (1996), Birkhäuser Verlag) · Zbl 0859.60001
[4] Carmona, P.; Petit, F.; Pitman, J.; Yor, M., On the laws of homogeneous functionals of the Brownian bridge, Studia Sci. Math. Hungar., 35, 445-455, (1999) · Zbl 0980.60099
[5] Csáki, E., On some distributions concerning maximum and minimum of a Wiener process, (Analytic Function Methods in Probability Theory (Proc. Colloq. Methods of Complex Anal. in the Theory of Probab. and Statist., Kossuth L. Univ. Debrecen, Debrecen, 1977), Colloq. Math. Soc. János Bolyai, vol. 21, (1979), North-Holland Amsterdam), 43-52
[6] Dudley, R. M., (Probabilities and Metrics: Convergence of Laws on Metric Spaces, with A View to Statistical Testing, Lecture Notes Series, vol. 45, (1976), Matematisk Institut, Aarhus Universitet Aarhus) · Zbl 0355.60004
[7] Dynkin, E. B., Some limit theorems for sums of independent random variables with infinite mathematical expectations, (Select. Transl. Math. Statist. and Probability, Vol. 1, (1961), Inst. Math. Statist. and Amer. Math. Soc. Providence, RI), 171-189 · Zbl 0112.10105
[8] Feller, W., The asymptotic distribution of the range of sums of independent random variables, Ann. Math. Statist., 22, 427-432, (1951) · Zbl 0043.34201
[9] Govindarajulu, Z.; Klotz, J. H., A note on the asymptotic distribution of the one-sample Kolmogorov-Smirnov statistic, Amer. Statist., 27, 164-165, (1973)
[10] Gradshteyn, I. S.; Ryzhik, I. M., Table of integrals, series, and products, (1994), Academic Press Inc. Boston, MA, Translation edited and with a preface by Alan Jeffrey · Zbl 0918.65002
[11] Kingman, J. F.C., (Poisson Processes, Oxford Science Publications, vol. 3, (1993), The Clarendon Press, Oxford University Press New York) · Zbl 0771.60001
[12] Kolmogorov, A. N., Sulla determinazione empirica di una legge di distribuzione, Giorn. Ist. Ital. Attuari, 4, 83-91, (1933) · Zbl 0006.17402
[13] Kosorok, M. R.; Lin, C.-Y., The versatility of function-indexed weight-rank statistics, J. Amer. Statist. Assoc., 94, 320-332, (1999) · Zbl 1072.62575
[14] Kuiper, N. H., Tests concering random points on a circle, Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math., 22, 38-47, (1960) · Zbl 0096.12504
[15] Lévy, P., Processus stochastiques et mouvement brownien. suivi d’une note de M. Loève, (1948), Gauthier-Villars Paris · Zbl 0034.22603
[16] Oberhettinger, F.; Badii, L., Tables of Laplace transforms, (1973), Springer-Verlag New York · Zbl 0285.65079
[17] Perman, M.; Wellner, J. A., On the distribution of Brownian areas, Ann. Appl. Probab., 6, 1091-1111, (1996) · Zbl 0870.60035
[18] Pitman, J.; Yor, M., Random Brownian scaling identities and splicing of Bessel processes, Ann. Probab., 26, 1683-1702, (1998) · Zbl 0937.60079
[19] Pitman, J.; Yor, M., Path decompositions of a Brownian bridge related to the ratio of its maximum and amplitude, Studia Sci. Math. Hungar., 35, 457-474, (1999) · Zbl 0973.60082
[20] Revuz, D.; Yor, M., (Continuous Martingales and Brownian Motion, Grundlehren der Mathematischen Wissenschaften, vol. 293, (1999), Springer-Verlag Berlin)
[21] Rogers, L. C.G.; Williams, D., (Diffusions, Markov Processes, and Martingales: Itô Calculus. Vol. 2, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, (1987), John Wiley & Sons Inc. New York)
[22] Salminen, P.; Vallois, P., On maximum increase and decrease of Brownian motion, Ann. Inst. H. Poincaré, 43, 655-676, (2007) · Zbl 1173.60338
[23] Shorack, G. R.; Wellner, J. A., (Empirical Processes with Applications to Statistics, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, (1986), John Wiley & Sons Inc. New York)
[24] Vervaat, W., A relation between Brownian bridge and Brownian excursion, Ann. Probab., 7, 143-149, (1979) · Zbl 0392.60058
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