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Various mixing properties of β -, β ′ - and Gaussian-Delaunay tessellations in ℝ d − 1 are studied. It is shown that these tessellation models are absolutely regular, or β -mixing. In the β - and the Gaussian case exponential bounds for the absolute regularity coefficients are found. In the β ′ -case these coefficients show a polynomial decay only. In the background are new and strong concentration bounds on the radius of stabilization of the underlying construction. Using a general device for absolutely regular stationary random tessellations, central limit theorems for a number of geometric parameters of β - and Gaussian-Delaunay tessellations are established. This includes the number of k -dimensional faces and the k -volume of the k -skeleton for k ∈ { 0 , 1 , ... , d − 1 }. [ABSTRACT FROM AUTHOR]

Let ξ 1 , ξ 2 , ... be a sequence of independent copies of a random vector in R d having an absolutely continuous distribution. Consider a random walk S i : = ξ 1 + ⋯ + ξ i , and let C n , d : = conv (0 , S 1 , S 2 , ... , S n) be the convex hull of the first n + 1 points it has visited. The polytope C n , d is called k-neighborly if for any indices 0 ≤ i 1 < ⋯ < i k ≤ n the convex hull of the k points S i 1 , ... , S i k is a (k - 1) -dimensional face of C n , d . We study the probability that C n , d is k-neighborly in various high-dimensional asymptotic regimes, i.e. when n, d, and possibly also k diverge to ∞ . There is an explicit formula for the expected number of (k - 1) -dimensional faces of C n , d which involves Stirling numbers of both kinds. Motivated by this formula, we introduce a distribution, called the Lah distribution, and study its properties. In particular, we provide a combinatorial interpretation of the Lah distribution in terms of random compositions and records, and explicitly compute its factorial moments. Limit theorems which we prove for the Lah distribution imply neighborliness properties of C n , d . This yields a new class of random polytopes exhibiting phase transitions parallel to those discovered by Vershik and Sporyshev, Donoho and Tanner for random projections of regular simplices and crosspolytopes. [ABSTRACT FROM AUTHOR]

In this paper, we prove three fundamental types of limit theorems for the q -norm of random vectors chosen at random in an ℓ p n -ball in high dimensions. We obtain a central limit theorem, a moderate deviations as well as a large deviations principle when the underlying distribution of the random vectors belongs to a general class introduced by Barthe, Guédon, Mendelson, and Naor. It includes the normalized volume and the cone probability measure as well as projections of these measures as special cases. Two new applications to random and non-random projections of ℓ p n -balls to lower-dimensional subspaces are discussed as well. The text is a continuation of [Z. Kabluchko, J. Prochno and C. Thäle, High-dimensional limit theorems for random vectors in ℓ p n -balls, Commun. Contemp. Math.21(1) (2019) 1750092]. [ABSTRACT FROM AUTHOR]

Let ξ 0 , ξ 1 , ... be i.i.d. random variables with zero mean and unit variance. Consider a random Taylor series of the form f (z) = ∑ k = 0 ∞ ξ k c k z k , where c 0 , c 1 , ... is a real sequence such that c n 2 is regularly varying with index γ − 1 , where γ > 0. We prove that 𝔼 N [ 0 , 1 − 𝜖 ] ∼ γ 2 π | log 𝜖 | as 𝜀 ↓ 0 , where N [ 0 , r ] denotes the number of real zeroes of f in the interval [ 0 , r ]. [ABSTRACT FROM AUTHOR]

We introduce and study the r -Lah distribution whose definition involves r -Stirling numbers of both kinds. We compute its expectation and variance, show its log-concavity and prove limit theorems for this distribution. We use these results to prove threshold phenomena for convex cones generated by random walks and to analyze the probability of unique recovery of sparse monotone signals from linear measurements. [ABSTRACT FROM AUTHOR]

Abstract: Let be independent identically distributed (i.i.d.) random variables with , . Suppose that for all and some . Let and . We are interested in the limiting distribution of the multiscale scan statistic We prove that for an appropriate normalizing sequence , the random variable converges to the Gumbel extreme-value law . The behavior of depends strongly on the distribution of the ’s. We distinguish between four cases. In the superlogarithmic case we assume that for every . In this case, we show that the main contribution to comes from the intervals having length of order , , where and is the order of the first non-vanishing cumulant of (not counting the variance). In the logarithmic case we assume that the function attains its maximum at some unique point . In this case, we show that the main contribution to comes from the intervals of length , , where . In the sublogarithmic case we assume that the tail of is heavier than , for some . In this case, the main contribution to comes from the intervals of length and in fact, under regularity assumptions, from the intervals of length 1. In the remaining, fourth case, the ’s are Gaussian. This case has been studied earlier in the literature. The main contribution comes from intervals of length , . We argue that our results cover most interesting distributions with light tails. The proofs are based on the precise asymptotic estimates for large and moderate deviation probabilities for sums of i.i.d. random variables due to Cramér, Bahadur, Ranga Rao, Petrov and others, and a careful extreme value analysis of the random field of standardized increments by the double sum method. [Copyright &y& Elsevier]

We study random polytopes of the form [ X 1 , ... , X n ] defined as convex hulls of independent and identically distributed random points X 1 , ... , X n in R d with one of the following densities: f d , β (x) = c d , β (1 − ‖ x ‖ 2) β , ‖ x ‖ < 1 , (beta distribution, β > − 1) or f ˜ d , β (x) = c ˜ d , β (1 + ‖ x ‖ 2) − β , x ∈ R d , (beta' distribution, β > d / 2). Here, c d , β and c ˜ d , β are normalizing constants. This setting also includes the uniform distribution on the unit sphere and the standard normal distribution as limiting cases. We derive exact and asymptotic formulae for the expected number of k -faces of [ X 1 , ... , X n ] for arbitrary k ∈ { 0 , 1 , ... , d − 1 }. We prove that for any such k this expected number is strictly monotonically increasing with n. Also, we compute the expected internal and external angles of these polytopes at faces of every dimension and, more generally, the expected conic intrinsic volumes of their tangent cones. By passing to the large n limit in the beta' case, we compute the expected f -vector of the convex hull of Poisson point processes with power-law intensity function. Using convex duality, we derive exact formulae for the expected number of k -faces of the zero cell for a class of isotropic Poisson hyperplane tessellations in R d. This family includes the zero cell of a classical stationary and isotropic Poisson hyperplane tessellation and the typical cell of a stationary Poisson–Voronoi tessellation as special cases. In addition, we prove precise limit theorems for this f -vector in the high-dimensional regime, as d → ∞. Finally, we relate the d -dimensional beta and beta' distributions to the generalized Pareto distributions known in extreme-value theory. [ABSTRACT FROM AUTHOR]