The Annals of Probability

Sur une conjecture de D. G. Kendall concernant la cellule de Crofton du plan et sur sa contrepartie brownienne

André Goldman

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Abstract

In connection with a conjecture stated by D. G. Kendall in the forties, we describe the asymptotic behavior of the distribution function of the area of the planar Crofton cell. We deduce from this (in support of his conjecture) that expressed in terms of eigenvalues, the large Crofton cells are nearly circular. We obtain also the asymptotic behavior of the Laplace transform of the law of the perimeter of the convex hull of planar Brownian motion run until time 1. This last result implies that the small convex hulls of Brownian motion are nearly circular.

Article information

Source
Ann. Probab., Volume 26, Number 4 (1998), 1727-1750.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855880

Digital Object Identifier
doi:10.1214/aop/1022855880

Mathematical Reviews number (MathSciNet)
MR1675067

Zentralblatt MATH identifier
0936.60009

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60J65: Brownian motion [See also 58J65]
Secondary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 60F10: Large deviations

Keywords
Poisson line process Crofton cell random polygons perimeter of the convex hull of Brownian motion eigenvalues

Citation

Goldman, André. Sur une conjecture de D. G. Kendall concernant la cellule de Crofton du plan et sur sa contrepartie brownienne. Ann. Probab. 26 (1998), no. 4, 1727--1750. doi:10.1214/aop/1022855880. https://projecteuclid.org/euclid.aop/1022855880


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