Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T18:41:02.340Z Has data issue: false hasContentIssue false
  • English
  • Français

On higher-dimensional analogues of the arc-sine law

Published online by Cambridge University Press:  14 July 2016

N. H. Bingham  and
R. A. Doney
N. H. Bingham*
Affiliation:
Royal Holloway and Bedford New College
R. A. Doney*
Affiliation:
University of Manchester
*
∗ Postal address: Department of Mathematics, Royal Holloway and Bedford New College, Egham Hill, Egham, Surrey TW20 0EX, UK.
∗∗ Postal address: Statistical Laboratory, Department of Mathematics, University of Manchester, Manchester M13 9PL, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

The arc-sine laws form one of the cornerstones of classical one-dimensional fluctuation theory. In higher dimensions, knowledge of fluctuation theory remains a great deal less complete. Motivated by this, we consider higher-dimensional analogues of the classical arc-sine laws.

Keywords

FLUCTUATION THEORYHIGHER DIMENSIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 1 , March 1988 , pp. 120 - 131
Copyright
Copyright © Applied Probability Trust 1988 

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

Barndorff-Nielsen, O. and Baxter, G. (1963) Combinatorial lemma in higher dimensions. Trans. Amer. Math. Soc. 108, 313–325.Google Scholar
Baxter, G. (1961) A combinatorial lemma for complex numbers. Ann. Math. Statist. 32, 901–904.Google Scholar
Cartier, P. and Foata, D. (1969) Problèmes combinatoires de commutation et réarrangements. Lecture Notes in Mathematics 85, Springer-Verlag, Berlin.CrossRefGoogle Scholar
David, F. N. (1953) A note on the evaluation of the multivariate normal integral. Biometrika 40, 458–459.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications , Volume II, 2nd edn. Wiley, New York.Google Scholar
Greenwood, P. E. and Shaked, M. (1977) Fluctuations of random walks in Rd and storage systems. Adv. Appl. Prob. 9, 566–587.Google Scholar
Hobby, C. and Pyke, R. (1963) Combinatorial methods in multi-dimensional fluctuation theory. Ann. Math. Statist. 24, 402–404.Google Scholar
Ito, K. and Mckean, H. P. (1965) Diffusion Processes and their Sample Paths. Springer-Verlag, Berlin.Google Scholar
Karatzas, I. and Shreve, S. E. (1984) Trivariate density of Brownian motion, its local and occupation times, with applications to stochastic control. Ann. Prob. 12, 819–828.Google Scholar
Karatzas, I. and Shreve, S. E. (1986 +) A decomposition of the Brownian path. To appear.Google Scholar
Karlin, S. and Taylor, H. L. (1981) A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
Kendall, M. G. (1941) Proof of relations connected with the tetrachoric series and its generalisations. Biometrika 32, 196–198.Google Scholar
Kendall, M. G. and Stuart, A. (1977) The Advanced Theory of Statistics, Volume 1: Distribution Theory , 4th edn. Griffin, London.Google Scholar
Kingman, J. F. C. (1966) On the algebra of queues. J. Appl. Prob. 3, 285–326.Google Scholar
Levy, P. (1939) Sur certains processus stochastiques homogènes. Compositio Math. 7, 283–339.Google Scholar
Levy, P. (1965) Processus stochastiques et mouvement brownien. Gauthier-Villars, Paris.Google Scholar
Lothaire, M. (1983) Combinatorics on Words. Encyclopaedia of Mathematics and its Applications 17, Addison-Wesley, Reading, Mass. Google Scholar
Mcfadden, J. A. (1960) Two expansions for the quadrivariate normal integral. Biometrika 47, 325–333.Google Scholar
Mogulskii, A. A. and Pecherskii, E. A. (1977) On the first exit time out of a semigroup in Rm for a random walk. Theory Prob. Appl. 22, 818–825.CrossRefGoogle Scholar
Moran, P. A. P. (1956) The numerical evaluation of a class of integrals, I, II. Proc. Camb. Phil. Soc. 52, 230–233, 442–488.Google Scholar
Moran, P. A. P. (1986) Orthant probabilities and Gaussian Markov processes. J. Appl. Prob. 23A, 413–417.Google Scholar
Pitman, J. W. and Yor, M. (1986) Asymptotic laws of planar Brownian motion. Ann. Prob. 14, 733–779.Google Scholar
Shimura, M. (1985) Excursions in a cone for two-dimensional Brownian motion. J. Math. Kyoto Univ. 25, 433–443.Google Scholar
Shimura, M. (1986 +) A limit theorem for two-dimensional Brownian motion conditioned to stay in a cone. To appear.Google Scholar
Simons, G. (1983) Lévy's equivalence for Brownian motion. Statist. Prob. Letters 1, 203–206.Google Scholar
Simons, G. (1986) A trivariate version of Lévy's equivalence. Statist. Prob. Letters 4, 7–8.Google Scholar