Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T00:31:33.245Z Has data issue: false hasContentIssue false
  • English
  • Français

The thrown string: a Markov field approach

Published online by Cambridge University Press:  01 July 2016

Leon Willenborg
Leon Willenborg*
Affiliation:
Netherlands Central Bureau of Statistics
*
∗Postal address: Netherlands Central Bureau of Statistics, P.O. Box 4481, 6401CZ Heerlen, The Netherlands.
Get access Rights & Permissions [Opens in a new window]

Abstract

A static model is proposed for a string thrown at random on a horizontal plane. The curvature of such a string is described by a Markov field that is derived from the Brownian bridge.

Keywords

BROWNIAN BRIDGESTATISTICAL MECHANICSCHAIN MOLECULES
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 3 , September 1985 , pp. 607 - 622
Copyright
Copyright © Applied Probability Trust 1985 

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.)

Footnotes

The views expressed in this paper are those of the author and do not necessarily reflect the policies of the Netherlands Central Bureau of Statistics. This paper is based on the author’s master’s thesis written under the supervision of Professor W. Vervaat, and presented to the Department of Mathematics of the Catholic University of Nijmegen in September 1980.

References

Abramowitz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions. Dover, New York.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Clarke, L. E. (1971) How long is a piece of string? Math. Gazette 55, 404–407.Google Scholar
Flory, P. J. (1969) Statistical Mechanics of Chain Molecules. Interscience Publishers, New York.Google Scholar
Gihman, I. I. and Skorohod, A. V. (1974) The Theory of Stochastic Processes , I. Springer-Verlag, Berlin.Google Scholar
Hardy, G. H. and Rogosinski, W. W. (1950) Fourier Series 2nd edn. Cambridge University Press.Google Scholar
Kesten, H. (1976) Existence and uniqueness of countable one-dimensional Markov random fields. Ann. Prob. 4, 557–569.Google Scholar
Kingman, J. F. C. (1977) The thrown string: an unsolved problem. Unpublished manuscript: abstract in Adv. Appl. Prob. 9, 431.Google Scholar
Kingman, J. F. C. (1982) The thrown string (with discussion). J. R. Statist. Soc. B 44, 109–138.Google Scholar
Levy, P. (1954) Théorie de l’addition des variables aléatoires , 2nd edn. Gauthier-Villars, Paris.Google Scholar
Raftery, A. (1979) Un problème de ficelle. C.R. Acad. Sci. Paris 289, 703–705.Google Scholar
Spitzer, F. (1975) Phase transition in one-dimensional nearest neighbor systems. J. Functional Anal. 20, 240–255.Google Scholar
Struik, D. (1961) Lectures on Classical Differential Geometry. Addison-Wesley, Reading, Ma.Google Scholar
Synge, J. L. (1968) Letter to the editor. Math. Gazette 52, 165.Google Scholar
Synge, J. L. (1970) The problem of the thrown string. Math. Gazette 54, 250–260.Google Scholar
Volkenstein, M. V. (1963) Configurational Statistics of Polymeric Chains. Interscience Publishers, New York.Google Scholar
Watts, A. M. (1983) On the problem of a thrown string. Paper presented at the Computational Techniques and Applications Conference, University of Sydney, August 1983.Google Scholar
Wax, N., (Ed.) (1954) Selected Papers on Noise and Stochastic Processes. Dover, New York.Google Scholar