Richard A. Vitale

An alternate formulation of mean value for random geometric figures

In contrast to the traditional technique of averaging functionals of a random geometric figure, it is possible to consider an averaging process which treats the figure as a whole. The result in probabilistic terminology is an expectation, which is itself a figure. Theoretical foundations and various instances are presented in an informal survey.

Regular simplices and Gaussian samples

We show that if a suitable type of simplex inRn is randomly rotated and its vertices projected onto a fixed subspace, they are as a point set affine-equivalent to a Gaussian sample in that subspace. Consequently, affine-invariant statistics behave the same for both mechanisms. In particular, the facet behavior for the convex hull is the same, as observed by Affentranger and Schneider; other results of theirs are translated into new results for the convex hulls of Gaussian samples. We show conversely that the conditions on the vertices of the simplex are necessary for this equivalence. Similar results hold for randomorthogonal transformations.

The Brunn-Minkowski inequality for random sets

The Brunn-Minkowski inequality asserts a concavity feature of the volume functional under convex addition of sets. Among its applications has been Andersons treatment of multivariate densities. Here we present a generalization which interprets the inequality in terms of random sets. This provides a natural proof of Mudholkars generalized Anderson-type inequality.

Some comparisons for Gaussian processes

Extensions and variants are given for the well-known comparison principle for Gaussian processes based on ordering by pairwise distance.

Covariances of symmetric statistics

We examine the second-order structure arising when a symmetric function is evaluated over intersecting subsets of random variables. The original work of Hoeffding is updated with reference to later results. New representations and inequalities are presented for covariances and applied to U-statistics.

Intrinsic Volumes of the Brownian Motion Body

Motivated from Gaussian processes, we derive the intrinsic volumes of the infinite-dimensional Brownian motion body . The method is by discretization to a class of orthoschemes. Numerical support is offered for a conjecture of Sangwine-Yager, and another conjecture is offered on the rate of decay of intrinsic volume sequences.

A differential version of the Efron-Stein inequality: bounding the variance of a function of an infinitely divisible variable

Upon a suitable passage to the limit, the Efron-Stein inequality produces a general variance bound for an absolutely continuous function of an infinitely divisible variable. A necessary and sufficient condition for attainment of the bound is also given.

On the volume of parallel bodies: a probabilistic derivation of the Steiner formula

We give a proof of the Steiner formula based on the theory of random convex bodies. In particular, we make use of laws of large numbers for both random volumes and random convex bodies themselves

Approximation by mutually completely dependent processes

Abstract Any stochastic process indexed on the integers can be approximated arbitrarily well in the L∞ sense by a stochastic process in which each value is (essentially) a 1-1 image of any other value. If the original process has continuous one dimensional marginal distributions, then these can be replicated in the approximating process.

Majorization and Gaussian correlation

Using Schur majorization, we show Gaussian correlation inequalities for two classes of sets: Schur cylinders and barycentrically ordered sets. As a corollary, we deduce positive correlation for pairs of centered convex sets that share the symmetries of a regular simplex.