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The distribution of volume reductions induced by isotropic random projections

Part of: Global differential geometry Geometric probability and stochastic geometry Multivariate analysis

Published online by Cambridge University Press:  01 July 2016

Jørgen Nielsen
Jørgen Nielsen*
Affiliation:
Danish Institute of Agricultural Sciences
*
∗ Postal address: Danish Institute of Agricultural Sciences, Biometry Research Unit, Research Centre Foulum, P.O. Box 50, DK-8830 Tjele, Denmark. Email address: Jorgen.Nielsen@agrsci.dk
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Abstract

In this paper, isotropic random projections of d-sets in ℝn are studied, where a d-set is a subset of a d-dimensional affine subspace which satisfies certain regularity conditions. The squared volume reduction induced by the projection of a d-set onto an isotropic random p-subspace is shown to be distributed as a product of independent beta-distributed random variables, for d ≤ p. One of the proofs of this result uses Wilks' lambda distribution from multivariate normal theory. The result is related to Cauchy's and Crofton's formulae in stochastic geometry. In particular, it can be used to give a new and quite simple proof of one of the classical Crofton intersection formulae.

Keywords

Beta distributionCauchy's formulaCrofton's formulaisotropic random projectionsmultivariate normal theorystochastic geometryWilks' lambda distribution

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 53C65: Integral geometry 62H10: Distribution of statistics
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 4 , December 1999 , pp. 985 - 994
Copyright
Copyright © Applied Probability Trust 1999 

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