Second moments related to Poisson hyperplane tessellations

TL;DR

This paper investigates the variance of the vertex number in the typical cell of a stationary Poisson hyperplane tessellation, providing bounds and characterizations related to the directional distribution.

Contribution

It offers sharp bounds for the variance of the vertex number and characterizes when the maximum occurs, extending formulas to general directional distributions.

Findings

Maximum variance occurs with rotation-invariant distributions.

Provided elementary proof for second moment formulas.

Extended formulas to non-isotropic directional distributions.

Abstract

We consider the typical cell of a stationary Poisson hyperplane tessellation in d-dimensional Euclidean space. It is well known that the expected vertex number of the typical cell is independent of the directional distribution of the hyperplane process. We give sharp bounds for the variance of this vertex number, showing, in particular, that the maximum of the variance is attained if and only if the distribution of the process is rotation invariant with respect to a suitable scalar product. The employed representation of the second moment of the vertex number is a special case of formulas providing the covariance matrix for the random vector whose components are the total k-face contents of the typical cell. In the isotropic case, such formulas were first obtained by R.E. Miles. We give a more elementary proof and extend the formulas to general directional distributions.