Skip to main content

On higher-dimensional analogues of the arc-sine law

  • N. H. Bingham (a1) and R. A. Doney (a2)

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.

Corresponding author
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.
Hide All
Barndorff-Nielsen, O. and Baxter, G. (1963) Combinatorial lemma in higher dimensions. Trans. Amer. Math. Soc. 108, 313325.
Baxter, G. (1961) A combinatorial lemma for complex numbers. Ann. Math. Statist. 32, 901904.
Cartier, P. and Foata, D. (1969) Problèmes combinatoires de commutation et réarrangements. Lecture Notes in Mathematics 85, Springer-Verlag, Berlin.
David, F. N. (1953) A note on the evaluation of the multivariate normal integral. Biometrika 40, 458459.
Feller, W. (1971) An Introduction to Probability Theory and its Applications , Volume II, 2nd edn. Wiley, New York.
Greenwood, P. E. and Shaked, M. (1977) Fluctuations of random walks in Rd and storage systems. Adv. Appl. Prob. 9, 566587.
Hobby, C. and Pyke, R. (1963) Combinatorial methods in multi-dimensional fluctuation theory. Ann. Math. Statist. 24, 402404.
Ito, K. and Mckean, H. P. (1965) Diffusion Processes and their Sample Paths. Springer-Verlag, Berlin.
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, 819828.
Karatzas, I. and Shreve, S. E. (1986 +) A decomposition of the Brownian path. To appear.
Karlin, S. and Taylor, H. L. (1981) A Second Course in Stochastic Processes. Academic Press, New York.
Kendall, M. G. (1941) Proof of relations connected with the tetrachoric series and its generalisations. Biometrika 32, 196198.
Kendall, M. G. and Stuart, A. (1977) The Advanced Theory of Statistics, Volume 1: Distribution Theory , 4th edn. Griffin, London.
Kingman, J. F. C. (1966) On the algebra of queues. J. Appl. Prob. 3, 285326.
Levy, P. (1939) Sur certains processus stochastiques homogènes. Compositio Math. 7, 283339.
Levy, P. (1965) Processus stochastiques et mouvement brownien. Gauthier-Villars, Paris.
Lothaire, M. (1983) Combinatorics on Words. Encyclopaedia of Mathematics and its Applications 17, Addison-Wesley, Reading, Mass.
Mcfadden, J. A. (1960) Two expansions for the quadrivariate normal integral. Biometrika 47, 325333.
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, 818825.
Moran, P. A. P. (1956) The numerical evaluation of a class of integrals, I, II. Proc. Camb. Phil. Soc. 52, 230233, 442488.
Moran, P. A. P. (1986) Orthant probabilities and Gaussian Markov processes. J. Appl. Prob. 23A, 413417.
Pitman, J. W. and Yor, M. (1986) Asymptotic laws of planar Brownian motion. Ann. Prob. 14, 733779.
Shimura, M. (1985) Excursions in a cone for two-dimensional Brownian motion. J. Math. Kyoto Univ. 25, 433443.
Shimura, M. (1986 +) A limit theorem for two-dimensional Brownian motion conditioned to stay in a cone. To appear.
Simons, G. (1983) Lévy's equivalence for Brownian motion. Statist. Prob. Letters 1, 203206.
Simons, G. (1986) A trivariate version of Lévy's equivalence. Statist. Prob. Letters 4, 78.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 6 *
Loading metrics...

Abstract views

Total abstract views: 97 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th March 2018. This data will be updated every 24 hours.