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Le problème de Buffon–Synge pour une corde

  • C. Donati-Martin (a1)

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.

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Postal address: Mathématiques, case 64, Université de Provence, 3 place Victor Hugo, 13331 Marseille Cedex, France.

References

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[2] Donati-Martin, C. (1989) Transformation de Fourier et temps d'occupation browniens. A paraître.
[3] Gihman, I. I. and Skorohod, A. V. (1975) The Theory of Stochastic Processes, Vol. II. Springer-Verlag, Berlin.
[4] Kallianpur, G. and Robbins, H. (1953) Ergodic property of Brownian motion process. Proc. Nat. Acad. Sci. USA 39, 525533.
[5] Kasahara, Y. (1977) Limit theorems of occupation times for Markov processes. Publ. RIMS, Kyoto Univ. 12, 801818.
[6] Kasahara, Y. and Kotani, S. (1979) On limit processes for a class of additive functionals of recurrent diffusion processes. Z. Wahrscheinlichkeitsth. 49, 133153.
[7] Kingman, J. F. C. (1982) The thrown string. J. R. Statist. Soc. B 44, 109138.
[8] Knight, F. B. (1971) A Reduction of Continuous Square-Integrable Martingales to Brownian Motion. Lecture Notes in Mathematics 190, Springer-Verlag, Berlin.
[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.
[10] Raftery, A. (1979) Un problème de ficelle. C.R. Acad. Sci. Paris A 289, 703705.
[11] Stroock, D. W. and Varadhan, S. R. S. (1979) Multidimensional Diffusion Processes. Springer-Verlag, Berlin.
[12] Synge, J. L. (1968) Letter to the Editor. Math. Gazette 52, 165.
[13] Synge, J. L. (1970) The problem of the thrown string. Math. Gazette 54, 250260.
[14] Willenborg, L. (1985) The thrown string: a Markov field approach. Adv. Appl. Prob. 17, 607622.
[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), 309322.
[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.

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Le problème de Buffon–Synge pour une corde

  • C. Donati-Martin (a1)

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