Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-24T06:46:15.858Z Has data issue: false hasContentIssue false
  • English
  • Français

Random p-content of a p-parallelotope in Euclidean n-space

Part of: General convexity Global differential geometry Geometric probability and stochastic geometry Multivariate analysis Distribution theory

Published online by Cambridge University Press:  01 July 2016

A. M. Mathai
A. M. Mathai*
Affiliation:
McGill University
*
∗ Postal address: Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec, Canada, H3A 2K6. Email address: mathai@math.mcgill.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

Techniques currently available in the literature in dealing with problems in geometric probabilities seem to rely heavily on results from differential and integral geometry. This paper provides a radical departure in this respect. By using purely algebraic procedures and making use of some properties of Jacobians of matrix transformations and functions of matrix argument, the distributional aspects of the random p-content of a p-parallelotope in Euclidean n-space are studied. The common assumptions of independence and rotational invariance of the random points are relaxed and the exact distributions and arbitrary moments, not just integer moments, are derived in this article. General real matrix-variate families of distributions, whose special cases include the mulivariate Gaussian, a multivariate type-1 beta, a multivariate type-2 beta and spherically symmetric distributions, are considered.

Keywords

Random parallelotoperandom simplexdistributions of the volumesareas and distancesmatrix-variate gamma and matrix-variate beta distributed random points

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 62E15: Exact distribution theory
Secondary: 52A22: Random convex sets and integral geometry 53C65: Integral geometry 62H10: Distribution of statistics
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 2 , June 1999 , pp. 343 - 354
Copyright
Copyright © Applied Probability Trust 1999 

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

Coleman, R. (1969). Random path through convex bodies. J. Appl. Prob. 6, 430–441.CrossRefGoogle Scholar
Fang, K.-I. and Anderson, T. W. (1990). Statistical Inference in Elliptically Contoured and Related Distributions. Allerton, New York.Google Scholar
Fang, K.-T. and Zhang, Y.-T. (1990). Generalized Multivariate Analysis. Springer/Sciences Press, Berlin/Beijing.Google Scholar
Fang, K.-T., Kotz, S. and Ng, K. W. (1990). Symmetric Multivariate and Related Distributions. (Monograph on Statistics and Applied Probability.) Chapman and Hall, New York.CrossRefGoogle Scholar
Kendall, M. G. and Moran., P. A. P. (1963). Geometrical Probability. Griffin, London.Google Scholar
Kingman, J. F. C. (1969). Random secants of a convex body. J. Appl. Prob. 6, 660–672.Google Scholar
Mathai, A. M. (1982). On a conjecture in geometric probability regarding asymptotic normality of a random simplex. Ann. Prob. 10, 247–251.Google Scholar
Mathai, A. M. (1993). A Handbook of Generalized Special Functions for Statistical and Physical Sciences. OUP, Oxford.Google Scholar
Mathai, A. M. (1997). Jacobians of Matrix Transformations and Functions of Matrix Argument. World Scientific, New York.Google Scholar
Miles, R. E. (1971). Isotropic random simplices. Adv. Appl. Prob. 3, 353–382.Google Scholar
Ruben, H. (1979). The volume of an isotropic random parallelotope. J. Appl. Prob. 16, 84–94.CrossRefGoogle Scholar
Ruben, H. and Miles, R. E. (1980). A canonical decomposition of the probability measure of sets of isotropic random points in Rn . J. Mult. Anal. 10, 1–18.CrossRefGoogle Scholar
Santaló, L. A. (1976). Integral Geometry and Geometric Probability. Addison-Wesley, Reading, UK.Google Scholar
Solomon, H. (1978). Geometric Probability. SIAM, Philadelphia, PA.Google Scholar