Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-16T14:49:21.896Z Has data issue: false hasContentIssue false
  • English
  • Français

Le problème de Buffon–Synge pour une corde

Published online by Cambridge University Press:  01 July 2016

C. Donati-Martin
C. Donati-Martin*
Affiliation:
Université de Provence
*
∗Postal address: Mathématiques, case 64, Université de Provence, 3 place Victor Hugo, 13331 Marseille Cedex, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper deals with the thrown string problem posed by Synge [12]. The models studied are those of Willenborg [14] and Kingman [7]. The convergence results obtained are complementary to those of Kingman, although they are derived by quite different methods involving the use of martingales.

Keywords

BROWNIAN MOTIONORNSTEIN–UHLENBECK BRIDGEMARTINGALESSTRICT CONVERGENCEKOLMOGOROV CRITERIONBURKHOLDER–DAVIS-GUNDY INEQUALITIESLIMIT LAWS
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 2 , June 1990 , pp. 375 - 395
Copyright
Copyright © Applied Probability Trust 1990 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bibliographie

[1] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[2] Donati-Martin, C. (1989) Transformation de Fourier et temps d'occupation browniens. A paraître.Google Scholar
[3] Gihman, I. I. and Skorohod, A. V. (1975) The Theory of Stochastic Processes, Vol. II. Springer-Verlag, Berlin.Google Scholar
[4] Kallianpur, G. and Robbins, H. (1953) Ergodic property of Brownian motion process. Proc. Nat. Acad. Sci. USA 39, 525–533.CrossRefGoogle ScholarPubMed
[5] Kasahara, Y. (1977) Limit theorems of occupation times for Markov processes. Publ. RIMS, Kyoto Univ. 12, 801–818.CrossRefGoogle Scholar
[6] Kasahara, Y. and Kotani, S. (1979) On limit processes for a class of additive functionals of recurrent diffusion processes. Z. Wahrscheinlichkeitsth. 49, 133–153.CrossRefGoogle Scholar
[7] Kingman, J. F. C. (1982) The thrown string. J. R. Statist. Soc. B 44, 109–138.Google Scholar
[8] Knight, F. B. (1971) A Reduction of Continuous Square-Integrable Martingales to Brownian Motion. Lecture Notes in Mathematics 190, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[9] Papanicolaou, G. C., Strook, D. W. and Varadhan, S. R. S. (1977) Martingale approach to some limit theorems. Duke Univ. Math. Ser. III, Statistical Mechanics and Dynamical Systems. Google Scholar
[10] Raftery, A. (1979) Un problème de ficelle. C.R. Acad. Sci. Paris A 289, 703–705.Google Scholar
[11] Stroock, D. W. and Varadhan, S. R. S. (1979) Multidimensional Diffusion Processes. Springer-Verlag, Berlin.Google Scholar
[12] Synge, J. L. (1968) Letter to the Editor. Math. Gazette 52, 165.CrossRefGoogle Scholar
[13] Synge, J. L. (1970) The problem of the thrown string. Math. Gazette 54, 250–260.CrossRefGoogle Scholar
[14] Willenborg, L. (1985) The thrown string: a Markov field approach. Adv. Appl. Prob. 17, 607–622.CrossRefGoogle Scholar
[15] Yamada, T. (1986) On some limit theorems for occupation times for a one-dimensional Brownian motion and its continuous additive functionals locally of zero energy. J. Math. Kyoto Univ. 26 (2), 309–322.Google Scholar
[16] Yor, M. (1983) Le drap brownien comme limite en loi des temps locaux linéaires. Séminaire de Probabilités XVII. Lecture Notes in Mathematics 986, Springer-Verlag, Berlin.Google Scholar