Let K be a smooth convex set. The convex hull of independent random points in K is a random polytope. Central limit theorems for the volume and the number of i dimensional faces of random polytopes are proved as the number of random points tends to infinity. One essential step is to determine the precise asymptotic order of the occurring variances.

Stein’s method is used to obtain two theorems on multivariate normal approximation. Our main theorem, Theorem 1.2, provides a bound on the distance to normality for any nonnegative random vector. Theorem 1.2 requires multivariate size bias coupling, which we discuss in studying the approximation of distributions of sums of dependent random vectors. In the univariate case, we briefly illustrate this approach for certain sums of nonlinear functions of multivariate normal variables. As a second illustration, we show that the multivariate distribution counting the number of vertices with given degrees in certain random graphs is asymptotically multivariate normal and obtain a bound on the rate of convergence. Both examples demonstrate that this approach may be suitable for situations involving non-local dependence. We also present Theorem 1.4 for sums of vectors having a local type of dependence. We apply this theorem to obtain a multivariate normal approximation for the distribution of the random p-vector which counts the number of edges in a fixed graph both of whose vertices have the same given color when each vertex is colored by one of p colors independently. All normal approximation results presented here do not require an ordering of the summands related to the dependence structure. This is in contrast to hypotheses of classical central limit theorems and examples, which involve e.g., martingale, Markov chain, or various mixing assumptions. Keywords and phrases: Stein’s method, coupling, size bias, random graphs, multivariate central limit theorems.

Abstract. We introduce a new version of Stein’s method that reduces a large class of normal approximation problems to variance bounding exercises, thus making a connection between central limit theorems and concentration of measure. Unlike Skorokhod embeddings, the object whose variance has to be bounded has an explicit formula that makes it possible to carry out the program more easily. As an application, we derive a general CLT for functions that are obtained as combinations of many local contributions, where the definition of ‘local ’ itself depends on the data. Several examples are given, including the solution to a nearest-neighbor CLT problem posed by Peter Bickel. 1.

Abstract. We introduce a version of Stein’s method for proving concentration and moment inequalities in problems with dependence. Simple illustrative examples from combinatorics, physics, and mathematical statistics are provided. 1. Introduction and

Motivation: DNA structure plays an important role in a variety of biological processes. Different di- and trinucleotide scales have been proposed to capture various aspects of DNA structure including base stacking energy, propeller twist angle, protein deformability, bendability, and position preference. Yet, a general framework for the computational analysis and prediction of DNA structure is still lacking. Such a framework should in particular address the following issues: (1) construction of sequences with extremal properties; (2) quantitative evaluation of sequences with respect to a given genomic background; (3) automatic extraction of extremal sequences and profiles from genomic data bases; (4) distribution and asymptotic behavior as the length N of the sequences increases; and (5) complete analysis of correlations between scales. Results: We develop a general framework for sequence analysis based on additive scales, structural or other, that addresses all these issues. We show how to construct extremal sequences and calibrate scores for automatic genomic and data base extraction. We show that distributions rapidly converge to normality as N increases. Pairwise correlations between scales depend both on background distribution and sequence length and rapidly converge to an analytically predictable asymptotic value. For di- and trinucleotide scales, normal behavior and asymptotic correlation values are attained over a characteristic window length of about 10-15 bp. With a uniform background distribution, pairwise correlations between empirically-derived scales remain relatively small and roughly constant at all lengths, except for propeller twist and protein deformability which are positively correlated. There is a positive (resp. negative) correlation between din...

Statistics arising in geometric probability can often be expressed as sums of stabilizing functionals, that is functionals which satisfy a local dependence structure. In this note we show that stabilization leads to nearly optimal rates of convergence in the CLT for statistics such as total edge length and total number of edges of graphs in computational geometry and the total number of particles accepted in random sequential packing models. These rates also apply to the 1-dimensional marginals of the random measures associated with these statistics.

Abstract. Since the introduction of Stein’s method in the early 1970s, much research has been done in extending and strengthening it; however, there does not exist a version of Stein’s original method of exchangeable pairs for multivariate normal approximation. The aim of this article is to fill this void. We present two abstract normal approximation theorems using exchangeable pairs in multivariate contexts, one for situations in which the underlying symmetries are discrete, and one for situations involving continuous symmetry groups. We provide several illustrative examples, including a multivariate version of Hoeffding’s combinatorial central limit theorem and a treatment of projections of Haar measure on the orthogonal and unitary groups. 1.

We prove the central limit theorem for the volume and the f-vector of the random polytope Pn and the Poisson random polytope Πn in a fixed convex polytope P ⊂ IR d. Here Pn is the convex hull of n random points in P, and Πn is the convex hull of the intersection of a Poisson process X(n), of intensity n, with P. A general lower bound on the variance is also proved.

Abstract. Choose n random, independent points in R d according to the normal distribution. Their convex hull Kn is the Gaussian random polytope. We prove that the volume and the number of faces of Kn satisfy the central limit theorem, settling a well known conjecture in the field. 1. The main result Let Ψd = Ψ denote the standard normal distribution on R d, its density function is ψd = ψ = 1 exp{−x2 (2π) d/2 2} where x 2 = |x | 2 is the square of the Euclidean norm of x ∈ R d. We will use this notation only for d ≥ 2, for d = 1 the standard normal has density function with distribution Φ. φ = 1 exp{−x2

Abstract. We introduce a new version of Stein’s method that reduces a large class of normal approximation problems to variance bounding exercises, thus making a connection between central limit theorems and concentration of measure. Unlike Skorokhod embeddings, the object whose variance has to be bounded has an explicit formula that makes it possible to carry out the program more easily. As an application, we derive a general CLT for functions that are obtained as combinations of many local contributions, where the definition of ‘local ’ itself depends on the data. Several examples are given, including the solution to a nearest-neighbor CLT problem posed by Peter Bickel. 1.