The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Convex bodies: the brunn–minkowski theory

Let $$U_1,U_2,\ldots $$
 be random points sampled uniformly and independently from the d-dimensional upper half-sphere. We show that, as $$n\rightarrow \infty $$
 , the f-vector of the $$(d+1)$$
 -dimensional convex cone $$C_n$$
 generated by $$U_1,\ldots ,U_n$$
 weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the f-vector of $$C_n$$
 and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of $$C_n$$
 can be expressed through the expected f-vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of Barany et al. (Random Struct Algorithms 50(1):3–22, 2017. https://doi.org/10.1002/rsa.20644
 ). Our approach is based on the observation that the random cone $$C_n$$
 weakly converges, after a suitable rescaling, to a random cone whose intersection with the tangent hyperplane of the half-sphere at its north pole is the convex hull of the Poisson point process with power-law intensity function proportional to $$\Vert x\Vert ^{-(d+\gamma )}$$
 , where $$\gamma =1$$
 . We compute the expected number of facets, the expected intrinsic volumes and the expected T-functional of this random convex hull for arbitrary $$\gamma >0$$
 .

Cones generated by random points on half-spheres and convex hulls of Poisson point processes

Let X1,⋯,Xn be i.i.d. random points in the d‐dimensional Euclidean space sampled according to one of the following probability densities: fd,β(x)=const·1−∥x∥2β,∥x∥

Expected intrinsic volumes and facet numbers of random beta-polytopes

Beta polytopes and Poisson polyhedra: f-vectors and angles

Let $X_1,\ldots,X_n$ be independent random points that are distributed according to a probability measure on $\mathbb{R}^d$ and let $P_n$ be the random convex hull generated by $X_1,\ldots,X_n$ ($n\geq d+1$). Natural classes of probability distributions are characterized for which, by means of Blaschke-Petkantschin formulae from integral geometry, one can show that the mean facet number of $P_n$ is strictly monotonically increasing in $n$.

Monotonicity of facet numbers of random convex hulls

Let $X=\{x_1,\ldots,x_n\} \subset \mathbb R^d$ be an $n$-element point set in general position. For a $k$-element subset $\{x_{i_1},\ldots,x_{i_k}\} \subset X$ let the degree ${\rm deg}_k(x_{i_1},\ldots,x_{i_k})$ be the number of empty simplices $\{x_{i_1},\ldots,x_{i_{d+1}}\} \subset X$ containing no other point of $X$. The $k$-degree of the set $X$, denoted ${\rm deg}_k(X)$, is defined as the maximum degree over all $k$-element subset of $X$. 
We show that if $X$ is a random point set consisting of $n$ independently and uniformly chosen points from a compact set $K$ then ${\rm deg}_d(X)=\Theta(n)$, improving results previously obtained by Barany, Marckert and Reitzner [Many empty triangles have a common edge, Discrete Comput. Geom., 2013] and Temesvari [Moments of the maximal number of empty simplices of a random point set, Discrete Comput. Geom., 2018] and giving the correct order of magnitude with a significantly simpler proof. Furthermore, we investigate ${\rm deg}_k(X)$. In the case $k=1$ we prove that ${\rm deg}_1(X)=\Theta(n^{d-1})$.

pdf/stars-of-empty-simplices-3l56k2bq3b.pdf

Stars of Empty Simplices.

Let $U_1,U_2,\ldots$ be random points sampled uniformly and independently from the $d$-dimensional upper half-sphere. We show that, as $n\to\infty$, the $f$-vector of the $(d+1)$-dimensional convex cone $C_n$ generated by $U_1,\ldots,U_n$ weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the $f$-vector of $C_n$ and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of $C_n$ can be expressed through the expected $f$-vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of B\'ar\'any, Hug, Reitzner and Schneider [Random points in halfspheres, Rand. Struct. Alg., 2017]. Our approach is based on the observation that the random cone $C_n$ weakly converges, after a suitable rescaling, to a random cone whose intersection with the tangent hyperplane of the half-sphere at its north pole is the convex hull of the Poisson point process with power-law intensity function proportional to $\|x\|^{-(d+\gamma)}$, where $\gamma=1$. We compute the expected number of facets, the expected intrinsic volumes and the expected $T$-functional of this random convex hull for arbitrary $\gamma>0$.

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Cones generated by random points on half-spheres and convex hulls of Poisson point processes

Expected intrinsic volumes and facet numbers of random beta-polytopes

Monotonicity of facet numbers of random convex hulls

Stars of Empty Simplices.

Cones generated by random points on half-spheres and convex hulls of Poisson point processes