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 Electronic Journal of Probability > Vol. 5 (2000) > Paper 10 open journal systems 


The Laws of Chung and Hirsch for Cauchy's Principal Values Related to Brownian Local Times

Yueyun Hu, Universite Paris VI


Abstract
Two Chung-type and Hirsch-type laws are established to describe the liminf asymptotic behaviours of the Cauchy's principal values related to Brownian local times. These results are generalized to a class of Brownian additive functionals.


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Pages: 1-16

Published on: April 4, 2000
Bibliography
  1. Bertoin, J., On the Hilbert transform of the local times of a Lévy process. Bull. Sci. Math. 119 (1995) 147--156 Math. Review 96e:60143
  2. Bertoin, J., Lévy Processes. Cambridge Press, 1996. Math. Review 98e:60117
  3. Bertoin, J., Cauchy's principal value of local times of Lévy's processes with no negative jumps via continuous branching processes. Electronic J. Probab. 2 (1997) 1--12 Math. Review 99b:60120
  4. Biane, Ph. and Yor, M., Valeurs principales associées aux temps locaux browniens. Bull. Sci. Math. 111 (1987) 23--101. Math. Review 88g:60188
  5. Chaumont, L., Excursion normalisée, méandre et pont pour les processus de Lévy stables. Bull. Sci. Math. 121 (1997) 377--403. Math. Review 99a:60077
  6. Chung, K.L., Excursions in Brownian motion. Ark. Math. 14 (1976) 155--177 Math. Review 57:7791
  7. Csáki, E., Csörgo, M., Földes, A. and Shi, Z., Increment sizes of the principal value of Brownian local time. Probab. Th. Rel. Fields (to appear)
  8. Csáki, E., Csörgo, M., Földes, A. and Shi, Z., Path properties of Cauchy's principal values related to local time. Studia Sci. Math. Hungar (to appear)
  9. Csáki, E., Erdos, P. and Révész, P., On the length of the longest excursion. Z. Wahrsch. verw. Gebiete 68 (1985) 365--382. Math. Review 86f:60086
  10. Csáki, E., Shi, Z. and Yor, M., Fractional Brownian motions as ``higher-order" fractional derivatives of Brownian local times. Colloquium on Limit Theorems in Probability and Statistics (Balatonlelle, Hungary) 1999.
  11. Csörgo, M. and Révész, P., Strong Approximations in Probability and Statistics Academic Press, New York (1981) Math. Review 84d:60050
  12. Donsker, M.D. and Varadhan, S.R.S., On the laws of the iterated logarithm for local times. Comm. Pure Appl. Math. 30 (1977) 707--753. Math. Review 57:1667
  13. Erdos, P., On the law of the iterated logarithm. Ann. Math. 43 (1942) 419--436. Math. Review 4,16j
  14. Fitzsimmons, P.J. and Getoor, R.K., On the distribution of the Hilbert transform of the local time of a symmetric Lévy processes. Ann. Probab. 20 (1992) 1484--1497. Math. Review 93j:60112
  15. Fitzsimmons, P.J. and Getoor, R.K., Limit theorems and variation properties for fractional derivatives of local time of a stable process. Ann. Inst. Henri Poincaré 28 (1992) 311--333. Math. Review 93j:60105
  16. Getoor, R.K., First passage times for symmetric stable processes in space. Trans. Amer. Math. Soc. 101 (1961) 75--90 Math. Review 25:604
  17. Gruet, J.C. and Shi, Z., The occupation time of Brownian motion in a ball. J. Theoret. Probab. 9 (1996) 429--445 Math. Review 97e:60135
  18. Hu, Y. and Shi, Z., An iterated logarithm law for Cauchy's principal value of Brownian local times. Exponential functionals and principal values related to Brownian motion, (1997) 211--228 Bibl. Rev. Mat. Iberoamericana, Madrid. Math. Review 1648661
  19. Hu, Y. and Shi, Z., Shortest excursion lengths. Ann. Inst. Henri Poincaré 1 (1999) 103--120. Math. Review 99k:60206
  20. Imhof, J.P., Density factorizations for Brownian motion and the three-dimensional Bessel processes and applications. J. Appl. Probab. 21 (1984) 500--510. Math. Review 85j:60152
  21. Jeanblanc, M., Pitman, J. and Yor, M., The Feynman-Kac formula and decomposition of Brownian paths. Mat. Apl. Comput. 16 (1997) 27--52. Math. Review 98f:60156
  22. Khoshnevisan, D. and Shi, Z., Chung's law for integrated Brownian motion. Trans. Amer. Math. Soc. 350 (1998) 4253--4264. Math. Review 98m:60056
  23. Kochen, S.B. and Stone, C.J., A note on the Borel--Cantelli lemma. Illinois J. Math. 8 (1964) 248--251. Math. Review 28:4562
  24. Kolmogorov, A.N., Sulla determinazione empirica delle leggi di probabilita. Giorn. Ist. Ital. Attuari 4 (1933) 1--11.
  25. Lamperti, J., An occupation time theorem for a class of stochastic processes. Trans. Amer. Math. Soc. 88 (1958) 380--387. Math. Review 20:1372
  26. Pitman, J.W. and Yor, M., A decomposition of Bessel bridges. Z. Wahrsch. verw. Gebiete 59 (1982) 425--457. Math. Review 84a:60091
  27. Pitman, J.W. and Yor, M., Arcsine laws and interval partitions derived from a stable subordinator. Proc. London Math. Soc. 65 (1992) 326--356. Math. Review 93e:60152
  28. Pitman, J.W. and Yor, M., The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 (1997) 855--900. Math. Review 98f:60147
  29. Revuz, D. and Yor, M., Continuous Martingales and Brownian Motion. (2nd edition) Springer, Berlin, 1994. Math. Review 95h:60072
  30. Shiryaev, A.N., Probability. (2nd edition) Springer, Berlin, 1996. Math. Review 97c:60003
  31. Shorack, G.R. and Wellner, J.A., Empirical Processes with Applications to Statistics. Wiley, New York, 1986. Math. Review 88e:60002
  32. Smirnov, N.V., On the estimation of the discrepancy between empirical curves of distribution for two independent samples. Bull. Math. Univ. Moscou 2 (1939) 3--14. (in Russian)
  33. Strassen, V., An invariance principle for the law of the iterated logarithm. Z. Wahrsch. verw. Gebiete. 3 (1964) 211--226. Math. Review 30:5379
  34. Yamada, T., Principal values of Brownian local times and their related topics. It^{o}'s Stochastic Calculus and Probability Theory (eds.: N. Ikeda, S.Watanabe, M. Fukushima, H. Kunita) pp. 413--422, Springer, 1996. Math. Review 98f:60160
  35. Yor, M., Some Aspects of Brownian Motion. Part I (Some Special Functionals). Lectures in Math. ETH Z"urich, Birkh"{a}user, Berlin, 1992. Math. Review 93j:60155
  36. Yor, M. (editor), Exponential Functionals and Principal Values Related to Brownian Motion. Biblioteca de la Revista Matem'atica Iberoamericana, Madrid, 1997. Math. Review 99d:60003
















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Electronic Journal of Probability. ISSN: 1083-6489