Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 10 (2005) > Paper 27 open journal systems 


On the Increments of the Principal Value of Brownian Local Time

Endre Csáki, Hungarian Academy of Sciences, Hungary
Yueyun Hu, Universite Paris VI


Abstract
Let $W$ be a one-dimensional Brownian motion starting from 0. Define $Y(t)= int_0^t{dsover W(s)}:= lim_{epsilonto 0} int_0^t 1_{(|W(s)|> epsilon)} {dsover W(s)}$ as Cauchy's principal value related to local time. We prove limsup and liminf results for the increments of $Y$.


Full text: PDF

Pages: 925-947

Published on: July 14, 2005
Bibliography
  1. Ait Ouahra, M. and Eddahbi, M.: Théorèmes limites pour certaines fonctionnelles associées aux processus stables sur l'espace de Hölder. Publ. Mat. 45 (2001), 371-386. 1876912
  2. Bertoin, J.: On the Hilbert transform of the local times of a Lévy process. Bull. Sci. Math. 119 (1995), 147-156. 1324841
  3. Bertoin, J.: Cauchy's principal value of local times of Lévy processes with no negative jumps via continuous branching processes. Electronic J. Probab. 2 (1997), Paper No. 6, 1-12. 1475864
  4. Biane, P. and Yor, M.: Valeurs principales associées aux temps locaux browniens. Bull. Sci. Math. 111 (1987), 23-101. 0886959
  5. Boufoussi, B., Eddahbi, M. and Kamont, A.: Sur la dérivée fractionnaire du temps local brownien. Probab. Math. Statist. 17 (1997), 311-319. 1490806
  6. Chung, K.L.: On the maximum partial sums of sequences of independent random variables. Trans. Amer. Math. Soc. 64 (1948), 205-233. 0026274
  7. Csáki, E., Csörgö, M. Földes, A. and Shi, Z.: Increment sizes of the principal value of Brownian local time. Probab. Th. Rel. Fields 117 (2000), 515-531. 1777131
  8. Csáki, E., Csörgö, M. Földes, A. and Shi, Z.: Path properties of Cauchy's principal values related to local time. Studia Sci. Math. Hungar. 38 (2001), 149-169. 1877775
  9. Csáki, E. and Földes, A.: A note on the stability of the local time of a Wiener process. Stoch. Process. Appl. 25 (1987), 203-213. 0915134
  10. Csáki, E., Földes, A. and Shi, Z.: A joint functional law for the Wiener process and principal value. Studia Sci. Math. Hungar. 40 (2003), 213-241. 2003004
  11. Csáki, E., Shi, Z. and Yor, M.: Fractional Brownian motions as "higher-order" fractional derivatives of Brownian local times. In: Limit Theorems in Probability and Statistics (I. Berkes et al., eds.) Vol. I, pp. 365-387. János Bolyai Mathematical Society, Budapest, 2002. 1979974
  12. Csörgö, M. and Révész, P.: Strong Approximations in Probability and Statistics. Academic Press, New York, 1981. 0666546
  13. Fitzsimmons, P.J. and Getoor, R.K.: On the distribution of the Hilbert transform of the local time of a symmetric Lévy process. Ann. Probab. 20 (1992), 1484-1497. 1175273
  14. Gradshteyn, I.S. and Ryzhik, I.M.: Table of Integrals, Series, and Products. Sixth ed. Academic Press, San Diego, CA, 2000. 1773820
  15. Grill, K.: On the last zero of a Wiener process. In: Mathematical Statistics and Probability Theory (M.L. Puri et al., eds.) Vol. A, pp. 99-104. D. Reidel, Dordrecht, 1987. 0922690
  16. Hu, Y.: The laws of Chung and Hirsch for Cauchy's principal values related to Brownian local times. Electronic J. Probab. 5 (2000), Paper No. 10, 1-16. 1768844
  17. Hu, Y. and Shi, Z.: An iterated logarithm law for Cauchy's principal value of Brownian local times. In: Exponential Functionals and Principal Values Related to Brownian Motion (M. Yor, ed.), pp. 131-154. Biblioteca de la Revista Matemática Iberoamericana, Madrid, 1997. 1648661
  18. Strassen, V.: An invariance principle for the law of the iterated logarithm. Z. Wahrsch. verw. Gebiete 3 (1964), 211-226. 0175194
  19. Wen, Jiwei: Some results on lag increments of the principal value of Brownian local time. Appl. Math. J. Chinese Univ. Ser. B 17 (2002), 199-207. 1904917
  20. Yamada, T.: Principal values of Brownian local times and their related topics. In: Itô's Stochastic Calculus and Probability Theory (N. Ikeda et al., eds.), pp. 413-422. Springer, Tokyo, 1996. 1439540
  21. Yor, M.: Some Aspects of Brownian Motion. Part 1: Some Special Functionals. ETH Zürich Lectures in Mathematics. Birkhäuser, Basel, 1992. 1193919
  22. Yor, M., editor: Exponential Functionals and Principal Values Related to Brownian Motion. Biblioteca de la Revista Matemática Iberoamericana, Madrid, 1997. 1648653
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489