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We study divisibility properties of a set { f 1 (U n (s)) , ... , f m (U n (s)) } , where f 1 , ... , f m are polynomials in s variables over Z and U n (s) is a point picked uniformly at random from the set { 1 , ... , n } s. We show that, as n → ∞ , the GCD and the suitably normalized LCM of this set converge in distribution to a.s. finite random variables under mild assumptions on f 1 , ... , f m. Our approach is based on the known fact that the uniform distribution on { 1 , ... , n } converges to the Haar measure on the ring Z ˆ of profinite integers, combined with the Lang–Weil bounds and tools from probability theory. [ABSTRACT FROM AUTHOR]

Consider a finite collection of affine hyperplanes in R d . The hyperplanes dissect R d into finitely many polyhedral chambers. For a point x ∈ R d and a chamber P the metric projection of x onto P is the unique point y ∈ P minimizing the Euclidean distance to x. The metric projection is contained in the relative interior of a uniquely defined face of P whose dimension is denoted by dim (x , P) . We prove that for every given k ∈ { 0 , ... , d } , the number of chambers P for which dim (x , P) = k does not depend on the choice of x, with an exception of some Lebesgue null set. Moreover, this number is equal to the absolute value of the k-th coefficient of the characteristic polynomial of the hyperplane arrangement. In a special case of reflection arrangements, this proves a conjecture of Drton and Klivans [A geometric interpretation of the characteristic polynomial of reflection arrangements. Proc. Amer. Math. Soc. 138(8), 2873–2887 (2010)]. [ABSTRACT FROM AUTHOR]

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 \mathcal Z_d be the zero cell of a d-dimensional, isotropic and stationary Poisson hyperplane tessellation. We study the asymptotic behavior of the expected number of k-dimensional faces of \mathcal Z_d, as d\to \infty. For example, we show that the expected number of hyperfaces of \mathcal Z_d is asymptotically equivalent to \sqrt {2\pi /3}\, d^{3/2}, as d\to \infty. We also prove that the expected solid angle of a random cone spanned by d random vectors that are independent and uniformly distributed on the unit upper half-sphere in \mathbb R^{d} is asymptotic to \sqrt 3 \pi ^{-d}, as d\to \infty. [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]

We consider the simplices K n A = { x ∈ R n + 1 : x 1 ≥ x 2 ≥ ⋯ ≥ x n + 1 , x 1 - x n + 1 ≤ 1 , x 1 + ⋯ + x n + 1 = 0 } and K n B = { x ∈ R n : 1 ≥ x 1 ≥ x 2 ≥ ⋯ ≥ x n ≥ 0 } , which are called the Schläfli orthoschemes of types A and B, respectively. We describe the tangent cones at their j-faces and compute explicitly the sums of the conic intrinsic volumes of these tangent cones at all j-faces of K n A and K n B . This setting contains sums of external and internal angles of K n A and K n B as special cases. The sums are evaluated in terms of Stirling numbers of both kinds. We generalize these results to finite products of Schläfli orthoschemes of type A and B and, as a probabilistic consequence, derive formulas for the expected number of j-faces of the Minkowski sums of the convex hulls of a finite number of Gaussian random walks and random bridges. Furthermore, we evaluate the analogous angle sums for the tangent cones of Weyl chambers of types A and B and finite products thereof. [ABSTRACT FROM AUTHOR]

We study random convex cones defined as positive hulls of d -dimensional random walks and bridges. We compute expectations of various geometric functionals of these cones such as the number of k -dimensional faces and the sums of conic quermassintegrals of their k -dimensional faces. These expectations are expressed in terms of Stirling numbers of both kinds and their B -analogues. [ABSTRACT FROM AUTHOR]

We study three families of polyhedral cones whose sections are regular simplices, cubes, and crosspolytopes. We compute solid angles and conic intrinsic volumes of these cones. We show that several quantities appearing in stochastic geometry can be expressed through these conic intrinsic volumes. A list of such quantities includes internal and external solid angles of regular simplices and crosspolytopes, the probability that a (symmetric) Gaussian random polytope or the Gaussian zonotope contains a given point, the expected number of faces of the intersection of a regular polytope with a random linear subspace passing through its centre, and the expected number of faces of the projection of a regular polytope onto a random linear subspace. [ABSTRACT FROM AUTHOR]

We determine the true asymptotic behaviour for the expected number of connected components for a model of random lemniscates proposed recently by Lerario and Lundberg. These are defined as the subsets of the Riemann sphere, where the absolute value of certain random, |$\textrm{SO}(3)$| -invariant rational function of degree |$n$| equals to |$1$|⁠. We show that the expected number of the connected components of these lemniscates, divided by |$n$|⁠ , converges to a positive constant defined in terms of the quotient of two independent plane Gaussian analytic functions. A major obstacle in applying the novel non-local techniques due to Nazarov and Sodin on this problem is the underlying non-Gaussianity, intrinsic to the studied model. [ABSTRACT FROM AUTHOR]

We continue our analysis of Ising models on the (directed) Erdős-Rényi random graph G(N, p). We prove a quenched Central Limit Theorem for the magnetization and describe the fluctuations of the log-partition function. In the current note we consider the low temperature regime β > 1 and the case when an external magnetic field is present. In both cases, we assume that p = p(N) satisfies p³N → ∞. [ABSTRACT FROM AUTHOR]

The β-Delaunay tessellation in ℝd-1 is a generalization of the classical Poisson-Delaunay tessellation. As a first result of this paper we show that the shape of a weighted typical cell of a βDelaunay tessellation, conditioned on having large volume, is close to the shape of a regular simplex inℝd-1. This generalizes earlier results of Hug and Schneider about the typical (non-weighted) Poisson-Delaunay simplex. Second, the asymptotic behaviour of the volume of weighted typical cells in high-dimensional β-Delaunay tessellation is analysed, as d → ∞. In particular, various high dimensional limit theorems, such as quantitative central limit theorems as well as moderate and large deviation principles, are derived. [ABSTRACT FROM AUTHOR]

The typical cell of a Voronoi tessellation generated by n + 1 uniformly distributed random points on the d-dimensional unit sphere S d is studied. Its f-vector is identified in distribution with the f-vector of a beta' polytope generated by n random points in R d . Explicit formulas for the expected f-vector are provided for any d and the low-dimensional cases d ∈ { 2 , 3 , 4 } are studied separately. This implies an explicit formula for the total number of k-dimensional faces in the spherical Voronoi tessellation as well. [ABSTRACT FROM AUTHOR]

Consider a random simplex [ X 1 , ... , X n ] defined as the convex hull of independent identically distributed (i.i.d.) random points X 1 , ... , X n in R n - 1 with the following beta density: Let J n , k (β) be the expected internal angle of the simplex [ X 1 , ... , X n ] at its face [ X 1 , ... , X k ] . Define J ~ n , k (β) analogously for i.i.d. random points distributed according to the beta ′ density f ~ n - 1 , β (x) ∝ (1 + ‖ x ‖ 2 ) - β , x ∈ R n - 1 , β > (n - 1) / 2. We derive formulae for J n , k (β) and J ~ n , k (β) which make it possible to compute these quantities symbolically, in finitely many steps, for any integer or half-integer value of β . For J n , 1 (± 1 / 2) we even provide explicit formulae in terms of products of Gamma functions. We give applications of these results to two seemingly unrelated problems of stochastic geometry: (i) We compute explicitly the expected f-vectors of the typical Poisson–Voronoi cells in dimensions up to 10. (ii) Consider the random polytope K n , d : = [ U 1 , ... , U n ] where U 1 , ... , U n are i.i.d. random points sampled uniformly inside some d-dimensional convex body K with smooth boundary and unit volume. Reitzner (Adv. Math. 191(1), 178–208 (2005)) proved the existence of the limit of the normalised expected f-vector of K n , d : lim n → ∞ n - (d - 1) / (d + 1) E f (K n , d) = c d · Ω (K) , where Ω (K) is the affine surface area of K, and c d is an unknown vector not depending on K. We compute c d explicitly in dimensions up to d = 10 and also solve the analogous problem for random polytopes with vertices distributed uniformly on the sphere. [ABSTRACT FROM AUTHOR]

We consider d-dimensional random vectors Y1, . . . ,Yn that satisfy a mild general position assumption a.s. The hyperplanes (Yi−Yj)⊥ (1 ≤ i < j ≤ n) generate a conical tessellation of the Euclidean d-space which is closely related to the Weyl chambers of type An−1. We determine the number of cones in this tessellation and show that it is a.s. constant. For a random cone chosen uniformly at random from this random tessellation, we compute expectations of several geometric functionals. These include the face numbers, as well as the conic intrinsic volumes and the conical quermassintegrals. Under the additional assumption of exchangeability on Y1, . . . ,Yn, the same is done for the dual random cones which have the same distribution as the positive hull of Y1−Y2, . . . , Yn−1−Yn conditioned on the event that this positive hull is not equal to Rd. All these expectations turn out to be distribution-free. Similarly, we consider the conical tessellation induced by the hyperplanes (Yi + Yj)⊥ (1 ≤ i < j ≤ n), (Yi − Yj)⊥ (1 ≤ i < j ≤ n), Yi⊥ (1 ≤ i ≤ n). This tessellation is closely related to the Weyl chambers of type Bn. We compute the number of cones in this tessellation and the expectations of various geometric functionals for random cones drawn from this random tessellation. The main ingredient in the proofs is a connection between the number of faces of the tessellation and the number of faces of the Weyl chambers of the corresponding type that are intersected non-trivially by a certain linear subspace in general position. [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 continue our analysis of Ising models on the (directed) Erdős–Rényi random graph. This graph is constructed on N vertices and every edge has probability p to be present. These models were introduced and first studied by Bovier and Gayrard (1993 J. Stat. Phys. 72 643–64) and further analyzed by the authors in a previous note, in which we consider the case of p = p(N) satisfying p3N2 → ∞ and β < 1. In the current note we prove a quenched central limit theorem for the magnetization for p satisfying pN → ∞ in the high-temperature regime β < 1. We also show a non-standard central limit theorem for p4N3 → ∞ at the critical temperature β = 1. For p4N3 → 0 we obtain a Gaussian limiting distribution for the magnetization. Finally, in the critical regime p4N3 → c the limiting distribution for the magnetization contains a Gaussian component as well as a -term. Hence, at β = 1 we observe a phase transition in p for the fluctuations of the magnetization. [ABSTRACT FROM AUTHOR]

Consider a random d-dimensional simplex whose vertices are d + 1 random points sampled independently and uniformly from the unit sphere in Rd. We show that the expected sum of solid angles at the vertices of this random simplex equals 1/8 if d = 3 and 539/288π2 − 1/6 if d = 4. The angles are measured as proportions of the full solid angle which is normalized to be 1. Similar formulae are obtained if the vertices of the simplex are uniformly distributed in the unit ball. These results are special cases of general formulae for the expected angle sums of random beta simplices in dimensions 3 and 4. [ABSTRACT FROM AUTHOR]

Let U 1 , U 2 , ... be random points sampled uniformly and independently from the d-dimensional upper half-sphere. We show that, as n → ∞ , the f-vector of the (d + 1) -dimensional convex cone C n generated by U 1 , ... , 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 Bárány et al. (Random Struct Algorithms 50(1):3–22, 2017. 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 ‖ x ‖ - (d + γ) , where γ = 1 . We compute the expected number of facets, the expected intrinsic volumes and the expected T-functional of this random convex hull for arbitrary γ > 0 . [ABSTRACT FROM AUTHOR]

We analyze Ising/Curie–Weiss models on the (directed) Erdős–Rényi random graph on N vertices in which every edge is present with probability p. These models were introduced by Bovier and Gayrard (J Stat Phys 72(3–4):643–664, 1993). We prove a quenched Central Limit Theorem for the magnetization in the high-temperature regime β < 1 when p = p (N) satisfies p 3 N 2 → + ∞ . [ABSTRACT FROM AUTHOR]

Let (B (t)) t ∈ Θ with Θ = Z or Θ = R be a wide sense stationary process with discrete or continuous time. The classical linear prediction problem consists of finding an element in span ¯ { B (s) , s ≤ t } providing the best possible mean square approximation to the variable B (τ) with τ > t. In this article we investigate this and some other similar problems where, in addition to prediction quality, optimization takes into account other features of the objects we search for. One of the most motivating examples of this kind is an approximation of a stationary process B by a stationary differentiable process X taking into account the kinetic energy that X spends in its approximation efforts. [ABSTRACT FROM AUTHOR]

Let X1, . . ., Xn be a standard normal sample in Rd. We compute exactly the expected volume of the Gaussian polytope conv [X1, . . ., Xn], the symmetric Gaussian polytope conv [±X1, . . .,±Xn], and the Gaussian zonotope [0,X1] + · · · + [0,Xn] by exploiting their connection to the regular simplex, the regular cross-polytope, and the cube with the aid of Tsirelson’s formula. The expected volumes of these random polytopes are given by essentially the same expressions as the intrinsic volumes and external angles of the regular polytopes. For all of these quantities, we obtain asymptotic formulae which are more precise than the results which were known before. More generally, we determine the expected volumes of some heteroscedastic random polytopes including conv [l1X1, . . ., lnXn] and conv[±l1X1, . . .,±lnXn], where l1, . . ., ln ≥ 0 are parameters, and the intrinsic volumes of the corresponding deterministic polytopes. Finally, we relate the kth intrinsic volume of the regular simplex Δn−1 to the expected maximum of independent standard Gaussian random variables ξ1, . . ., ξn given that the maximum has multiplicity k. Namely, we show that Vk(Δn−1) = (2π)k/2/k! · lim ε↓0 ε1−kE[max{ξ1, . . ., ξn}1{ξ(n) −ξ(n−k+1) ≤ε}], where ξ(1) ≤ ·· · ≤ ξ(n) denote the order statistics. A similar result holds for the cross-polytope if we replace ξ1, . . ., ξn with their absolute values. [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]

In this paper, we prove a multivariate central limit theorem for ℓ q -norms of high-dimensional random vectors that are chosen uniformly at random in an ℓ p n -ball. As a consequence, we provide several applications on the intersections of ℓ p n -balls in the flavor of Schechtman and Schmuckenschläger and obtain a central limit theorem for the length of a projection of an ℓ p n -ball onto a line spanned by a random direction 𝜃 ∈ 𝕊 n − 1 . The latter generalizes results obtained for the cube by Paouris, Pivovarov and Zinn and by Kabluchko, Litvak and Zaporozhets. Moreover, we complement our central limit theorems by providing a complete description of the large deviation behavior, which covers fluctuations far beyond the Gaussian scale. In the regime 1 ≤ p < q this displays in speed and rate function deviations of the q -norm on an ℓ p n -ball obtained by Schechtman and Zinn, but we obtain explicit constants. [ABSTRACT FROM AUTHOR]

Let r = r(n) be a sequence of integers such that r ≤ n and let X1, ..., Xr+1 be independent random points distributed according to the Gaussian, the Beta or the spherical distribution on ℝn. Limit theorems for the log-volume and the volume of the random convex hull of X1, ..., Xr+1 are established in high dimensions, that is, as r and n tend to infinity simultaneously. This includes Berry-Esseen-type central limit theorems, log-normal limit theorems, and moderate and large deviations. Also different types of mod-φ convergence are derived. The results heavily depend on the asymptotic growth of r relative to n. For example, we prove that the fluctuations of the volume of the simplex are normal (respectively, log-normal) if r = o(n) (respectively, r ~ αn for some 0 < α < 1). [ABSTRACT FROM AUTHOR]

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∥<1,(thebetacase)and f∼d,β(x)=const·1+∥x∥2−β,x∈Rd,(thebeta"case).We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of X1,⋯,Xn. Asymptotic formulae were obtained previously by Affentranger [The convex hull of random points with spherically symmetric distributions, 1991]. By studying the limits of the beta case when β↓−1, respectively β↑+∞, we can also cover the models in which X1,⋯,Xn are uniformly distributed on the unit sphere or normally distributed, respectively. We obtain similar results for the random polytopes defined as the convex hulls of ±X1,⋯,±Xn and 0,X1,⋯,Xn. One of the main tools used in the proofs is the Blaschke–Petkantschin formula. [ABSTRACT FROM AUTHOR]

Let Pn be an n-dimensional regular polytope from one of the three infinite series (regular simplices, regular crosspolytopes, and cubes). Project Pn onto a random, uniformly distributed linear subspace of dimension d = 2. We prove that the expected number of k-dimensional faces of the resulting random polytope is an increasing function of n. As a corollary, we show that the expected number of k-faces of the Gaussian polytope is an increasing function of the number of points used to generate the polytope. Similar results are obtained for the symmetric Gaussian polytope and the Gaussian zonotope. [ABSTRACT FROM AUTHOR]

We prove weak convergence on the Skorokhod space of Galton–Watson processes with immigration, properly normalized, under the assumption that the tail of the immigration distribution has a logarithmic decay. The limits are extremal shot noise processes. By considering marginal distributions, we recover the results of Pakes (1979). [ABSTRACT FROM AUTHOR]

Consider a sequence of partial sums S i = ξ 1 + … + ξ i , 1 ≤ i ≤ n , starting at S 0 = 0 , whose increments ξ 1 , … , ξ n are random vectors in R d , d ≤ n . We are interested in the properties of the convex hull C n : = Conv ( S 0 , S 1 , … , S n ) . Assuming that the tuple ( ξ 1 , … , ξ n ) is exchangeable and a certain general position condition holds, we prove that the expected number of k -dimensional faces of C n is given by the formula E [ f k ( C n ) ] = 2 ⋅ k ! n ! ∑ l = 0 ∞ [ n + 1 d − 2 l ] { d − 2 l k + 1 } , for all 0 ≤ k ≤ d − 1 , where [ n m ] and { n m } are Stirling numbers of the first and second kind, respectively. Further, we compute explicitly the probability that for given indices 0 ≤ i 1 < … < i k + 1 ≤ n , the points S i 1 , … , S i k + 1 form a k -dimensional face of Conv ( S 0 , S 1 , … , S n ) . This is done in two different settings: for random walks with symmetrically exchangeable increments and for random bridges with exchangeable increments. These results generalize the classical one-dimensional discrete arcsine law for the position of the maximum due to E. Sparre Andersen. All our formulae are distribution-free, that is do not depend on the distribution of the increments ξ k 's. The main ingredient in the proof is the computation of the probability that the origin is absorbed by a joint convex hull of several random walks and bridges whose increments are invariant with respect to the action of direct product of finitely many reflection groups of types A n − 1 and B n . This probability, in turn, is related to the number of Weyl chambers of a product-type reflection group that are intersected by a linear subspace in general position. [ABSTRACT FROM AUTHOR]

Consider a branching random walk on Z in discrete time. Denote by Ln(k) the number of particles at site k ∈ Z at time n ∈ N0. By the profile of the branching random walk (at time n) we mean the function k → Ln(k). We establish the following asymptotic expansion of Ln(k), as n→∞: e-Φ(0)nLn(k) = e-1/2x2 n(k)√ 2πΦ (0)n Σr j=0 Fj (xn(k))/nj/2 +o(n -r+1/2 ) a.s., where r ∈ N0 is arbitrary, Φ(β) = logΣk∈Z eβkEL1(k) is the cumulant generating function of the intensity of the branching random walk and xn(k) = k -Φ (0)n/√Φ'' (0)n . The expansion is valid uniformly in k ∈ Z with probability 1 and the Fj's are polynomials whose random coefficients can be expressed through the derivatives of Φ and the derivatives of the limit of the Biggins martingale at 0. Using exponential tilting, we also establish more general expansions covering the whole range of the branching random walk except its extreme values. As an application of this expansion for r = 0, 1, 2 we recover in a unified way a number of known results and establish several new limit theorems. In particular, we study the a.s. behavior of the individual occupation numbers Ln(kn), where kn ∈ Z depends on n in some regular way. We also prove a.s. limit theorems for the mode argmaxk∈Z Ln(k) and the height maxk∈Z Ln(k) of the profile. The asymptotic behavior of these quantities depends on whether the drift parameter Φ'(0) is integer, non-integer rational, or irrational. Applications of our results to profiles of random trees including binary search trees and random recursive trees will be given in a separate paper. [ABSTRACT FROM AUTHOR]

Consider n points $$X_1,\ldots ,X_n$$ in $$\mathbb {R}^d$$ and denote their convex hull by $${\Pi }$$ . We prove a number of inclusion-exclusion identities for the system of convex hulls $${\Pi }_I:=\mathrm{conv}(X_i: i\in I)$$ , where I ranges over all subsets of $$\{1,\ldots ,n\}$$ . For instance, denoting by $$c_k(X)$$ the number of k-element subcollections of $$(X_1,\ldots ,X_n)$$ whose convex hull contains a point $$X\in \mathbb {R}^d$$ , we prove that for all X in the relative interior of $${\Pi }$$ . This confirms a conjecture of Cowan (Adv Appl Probab 39(3):630-644, 2007) who proved the above formula for almost all X. We establish similar results for the number of polytopes $${\Pi }_J$$ containing a given polytope $${\Pi }_I$$ as an r-dimensional face, thus proving another conjecture of Cowan (Discrete Comput Geom 43(2):209-220, 2010). As a consequence, we derive inclusion-exclusion identities for the intrinsic volumes and the face numbers of the polytopes $${\Pi }_I$$ . The main tool in our proofs is a formula for the alternating sum of the face numbers of a convex polytope intersected by an affine subspace. This formula generalizes the classical Euler-Schläfli-Poincaré relation and is of independent interest. [ABSTRACT FROM AUTHOR]

We give an explicit formula for the probability that the convex hull of an n-step random walk in $${\mathbb{R}^d}$$ does not contain the origin, under the assumption that the distribution of increments of the walk is centrally symmetric and puts no mass on affine hyperplanes. This extends the formula by Sparre Andersen (Skand Aktuarietidskr 32:27-36, 1949) for the probability that such random walk in dimension one stays positive. Our result is distribution-free, that is, the probability does not depend on the distribution of increments. This probabilistic problem is shown to be equivalent to either of the two geometric ones: (1) Find the number of Weyl chambers of type B intersected by a generic linear subspace of $${\mathbb{R}^n}$$ of codimension d; (2) Find the conic intrinsic volumes of a Weyl chamber of type B . We solve the first geometric problem using the theory of hyperplane arrangements. A by-product of our method is a new simple proof of the general formula by Klivans and Swartz (Discrete Comput Geom 46(3):417-426, 2011) relating the coefficients of the characteristic polynomial of a linear hyperplane arrangement to the conic intrinsic volumes of the chambers constituting its complement. We obtain analogous distribution-free results for Weyl chambers of type $${A_{n-1}}$$ (yielding the probability of absorption of the origin by the convex hull of a generic random walk bridge), type D , and direct products of Weyl chambers (yielding the absorption probability for the joint convex hull of several random walks or bridges). The simplest case of products of the form $${B_1\times \cdots \times B_1}$$ recovers the Wendel formula (Math Scand 11:109-111, 1962) for the probability that the convex hull of an i.i.d. multidimensional sample chosen from a centrally symmetric distribution does not contain the origin. We also give an asymptotic analysis of the obtained absorption probabilities as $${n \to \infty}$$ , in both cases of fixed and increasing dimension d. [ABSTRACT FROM AUTHOR]

We revisit conservative/dissipative and positive/null decompositions of stationary max-stable processes. Originally, both decompositions were defined in an abstract way based on the underlying non-singular flow representation. We provide simple criteria which allow to tell whether a given spectral function belongs to the conservative/dissipative or positive/null part of the de Haan spectral representation. Specifically, we prove that a spectral function is null-recurrent iff it converges to 0 in the Cesàro sense. For processes with locally bounded sample paths we show that a spectral function is dissipative iff it converges to 0 . Surprisingly, for such processes a spectral function is integrable a.s. iff it converges to 0 a.s. Based on these results, we provide new criteria for ergodicity, mixing, and existence of a mixed moving maximum representation of a stationary max-stable process in terms of its spectral functions. In particular, we study a decomposition of max-stable processes which characterizes the mixing property. [ABSTRACT FROM AUTHOR]

A fractionally integrated inverse stable subordinator (FIISS) is the convolution of an inverse stable subordinator, also known as a Mittag-Leffler process, and a power function. We show that the FIISS is a scaling limit in the Skorokhod space of a renewal shot noise process with heavy-tailed, infinite mean ‘inter-shot’ distribution and regularly varying response function. We prove local Hölder continuity of FIISS and a law of iterated logarithm for both small and large times. [ABSTRACT FROM AUTHOR]

A formula due to Sudakov relates the first intrinsic volume of a convex set in a Hilbert space to the maximum of the isonormal Gaussian process over this set. Using this formula we compute the first intrinsic volume of some infinite-dimensional convex compact sets including unit balls with respect to Sobolev-type seminorms and ellipsoids in the Hilbert space. We relate the distribution of the random one-dimensional projections of these sets to the distributions S1, S2, C1, C2 studied by Biane, Pitman, Yor [Bull. AMS 38 (2001)]. We show that the k-th intrinsic volume of the set of all functions on [0, 1] which have Lipschitz constant bounded by 1 and which vanish at 0 (respectively, which have vanishing integral) is given by ... This is related to the results of Gao and Vitale [Discrete Comput. Geom. 26 (2001); Elect. Comm. Probab. 8 (2003)], who considered a similar question for functions with a restriction on the total variation instead of the Lipschitz constant. Using the results of Gao and Vitale we give a new proof of the formula for the expected volume of the convex hull of the d-dimensional Brownian motion which is due to Eldan [Elect. J. Probab. 19 (2014)]. Additionally, we prove an analogue of Eldan's result for the Brownian bridge. Similarly, we show that the results on the intrinsic volumes of the Lipschitz balls can be translated into formulae for the expected volumes of zonoids (Aumann integrals) generated by the Brownian motion and the Brownian bridge. Also, these results have discrete versions for Gaussian random walks and bridges. Our proofs exploit Sudakov's and Tsirelson's theorems which establish a connection between the intrinsic volumes and the isonormal Gaussian process. [ABSTRACT FROM AUTHOR]

Motivated by problems of hyperbolic stochastic geometry we introduce and study the class of beta-star polytopes. A beta-star polytope is defined as the convex hull of an inhomogeneous Poisson processes on the complement of the unit ball in R d with density proportional to (‖ x ‖ 2 − 1) − β , where ‖ x ‖ > 1 and β > d / 2. Explicit formulas for various geometric and combinatorial functionals associated with beta-star polytopes are provided, including the expected number of k -dimensional faces, the expected external angle sums and the expected intrinsic volumes. Beta-star polytopes are relevant in the context of hyperbolic stochastic geometry, since they are tightly connected to the typical cell of a Poisson-Voronoi tessellation as well as the zero cell of a Poisson hyperplane tessellation in hyperbolic space. The general results for beta-star polytopes are used to provide explicit formulas for the expected f -vector of the typical hyperbolic Poisson-Voronoi cell and the hyperbolic Poisson zero cell. Their asymptotics for large intensities and their monotonicity behaviour is discussed as well. Finally, stochastic geometry in the de Sitter half-space is studied as the hyperbolic analogue to recent investigations about random cones generated by random points on half-spheres in spherical or conical stochastic geometry. [ABSTRACT FROM AUTHOR]

In the classical leader election procedure all players toss coins independently and those who get tails leave the game, while those who get heads move to the next round where the procedure is repeated. We investigate a generalizion of this procedure in which the labels (positions) of the players who remain in the game are determined using an integer-valued random walk. We study the asymptotics of some relevant quantities for this model such as: the positions of the persons who remained after n rounds; the total number of rounds until all the persons among 1, 2,..., M leave the game; and the number of players among 1, 2,..., M who survived the first n rounds. Our results lead to some interesting connection with Galton-Watson branching processes and with the solutions of certain stochasticfixed point equations arising in the context of the stability of point processes under thinning. We describe the set of solutions to these equations and thus provide a characterization of one-dimensional point processes that are stable with respect to thinning by integer-valued random walks. [ABSTRACT FROM AUTHOR]

We study stationary max-stable processes { η ( t ) : t ∈ R } admitting a representation of the form η ( t ) = max i ∈ N ( U i + Y i ( t ) ) , where ∑ i = 1 ∞ δ U i is a Poisson point process on R with intensity e − u d u , and Y 1 , Y 2 , … are i.i.d. copies of a process { Y ( t ) : t ∈ R } obtained by running a Lévy process for positive t and a dual Lévy process for negative t . We give a general construction of such Lévy–Brown–Resnick processes, where the restrictions of Y to the positive and negative half-axes are Lévy processes with random birth and killing times. We show that these max-stable processes appear as limits of suitably normalized pointwise maxima of the form M n ( t ) = max i = 1 , … , n ξ i ( s n + t ) , where ξ 1 , ξ 2 , … are i.i.d. Lévy processes and s n is a sequence such that s n ∼ c log n with c > 0 . Also, we consider maxima of the form max i = 1 , … , n Z i ( t / log n ) , where Z 1 , Z 2 , … are i.i.d. Ornstein–Uhlenbeck processes driven by an α -stable noise with skewness parameter β = − 1 . After a linear normalization, we again obtain limiting max-stable processes of the above form. This gives a generalization of the results of Brown and Resnick (1977) to the totally skewed α -stable case. [ABSTRACT FROM AUTHOR]

Let X1, X2,... be independent identically distributed random variables with values in C. Denote by µ the probability distribution of X1. Consider a random polynomial Pn(z) = (z -- X1)... (z -- Xn). We prove a conjecture of Pemantle and Rivin [in: I. Kotsireas and E. V. Zima, eds., Advances in Combinatorics, Waterloo Workshop in Computer Algebra, 2011] that the empirical measure ... counting the complex zeros of the derivative P'n converges in probability to µ, as n → ∞. [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]

The partition function of the random energy model at inverse temperature $$\beta $$ is a sum of random exponentials $$ \mathcal{Z }_N(\beta )=\sum _{k=1}^N \exp (\beta \sqrt{n} X_k)$$, where $$X_1,X_2,\ldots $$ are independent real standard normal random variables (=random energies), and $$n=\log N$$. We study the large N limit of the partition function viewed as an analytic function of the complex variable $$\beta $$. We identify the asymptotic structure of complex zeros of the partition function confirming and extending predictions made in the theoretical physics literature. We prove limit theorems for the random partition function at complex $$\beta $$, both on the logarithmic scale and on the level of limiting distributions. Our results cover also the case of the sums of independent identically distributed random exponentials with any given correlations between the real and imaginary parts of the random exponent. [ABSTRACT FROM AUTHOR]

Let (a_k)_{k\in \mathbb N} be an increasing sequence of positive integers satisfying the Hadamard gap condition a_{k+1}/a_k> q >1 for all k\in \mathbb N, and let \begin{equation*} S_n(\omega) = \sum _{k=1}^n \cos (2\pi a_k \omega), \qquad n\in \mathbb N, \; \omega \in [0,1]. \end{equation*} Then S_n is called a lacunary trigonometric sum, and can be viewed as a random variable defined on the probability space \Omega = [0,1] endowed with Lebesgue measure. Lacunary sums are known to exhibit several properties that are typical for sums of independent random variables. For example, a central limit theorem for (S_n)_{n\in \mathbb {N}} has been obtained by Salem and Zygmund, while a law of the iterated logarithm is due to Erdős and Gál. In this paper we study large deviation principles for lacunary sums. Specifically, under the large gap condition a_{k+1}/a_k \to \infty, we prove that the sequence (S_n/n)_{n \in \mathbb {N}} does indeed satisfy a large deviation principle with speed n and the same rate function \widetilde {I} as for sums of independent random variables with the arcsine distribution. On the other hand, we show that the large deviation principle may fail to hold when we only assume the Hadamard gap condition. However, we show that in the special case when a_k= q^k for some q\in \{2,3,\ldots \}, (S_n/n)_{n \in \mathbb {N}} satisfies a large deviation principle (with speed n) and a rate function I_q that is different from \widetilde {I}, and describe an algorithm to compute an arbitrary number of terms in the Taylor expansion of I_q. In addition, we also prove that I_q converges pointwise to \widetilde I as q\to \infty. Furthermore, we construct a random perturbation (a_k)_{k \in \mathbb {N}} of the sequence (2^k)_{k \in \mathbb {N}} for which a_{k+1}/a_k \to 2 as k\to \infty, but for which at the same time (S_n/n)_{n \in \mathbb {N}} satisfies a large deviation principle with the same rate function \widetilde {I} as in the independent case, which is surprisingly different from the rate function I_2 one might naïvely expect. We relate this fact to the number of solutions of certain Diophantine equations. Together, these results show that large deviation principles for lacunary trigonometric sums are very sensitive to the arithmetic properties of the sequence (a_k)_{k\in \mathbb N}. This is particularly noteworthy since no such arithmetic effects are visible in the central limit theorem or in the law of the iterated logarithm for lacunary trigonometric sums. Our proofs use a combination of tools from probability theory, harmonic analysis, and dynamical systems. [ABSTRACT FROM AUTHOR]

Abstract: Let be a -dimensional array of independent standard Gaussian random variables. For a finite set define . Let be the number of elements in . We prove that the appropriately normalized maximum of , where ranges over all discrete cubes or rectangles contained in , converges in law to the Gumbel extreme-value distribution as . We also prove a continuous-time counterpart of this result. [ABSTRACT FROM AUTHOR]

Abstract: We prove that a stationary max-infinitely divisible process is mixing (ergodic) iff its dependence function converges to 0 (is Cesàro summable to 0). These criteria are applied to some classes of max-infinitely divisible processes. [Copyright &y& Elsevier]

Abstract: Let be a space–time Gaussian process which is stationary in the time variable . We study , the supremum of taken over and rescaled by a properly chosen sequence . Under appropriate conditions on , we show that for some normalizing sequence , the process converges as to a stationary max-stable process of Brown–Resnick type. Using strong approximation, we derive an analogous result for the empirical process. [Copyright &y& Elsevier]

We generalize a theorem of Shao [Proc. Amer. Math. Soc. 123 (1995) 575-582] on the almost-sure limiting behavior of the maximum of standardized random walk increments to multidimensional arrays of i.i.d. random variables. The main difficulty is the absence of an appropriate strong approximation result in the multidimensional setting. The multiscale statistic under consideration was used recently for the selection of the regularization parameter in a number of statistical algorithms as well as for the multiscale signal detection. [ABSTRACT FROM AUTHOR]

Pick d + 1 points uniformly at random on the unit sphere in R d. What is the expected value of the angle sum of the simplex spanned by these points? Choose n points uniformly at random in the d -dimensional ball. What is the expected number of faces of their convex hull? We answer these and some other, seemingly unrelated, questions of stochastic geometry. To this end, we compute expected internal angles of random simplices whose vertices are independent random points sampled from one of the following d -dimensional distributions: (i) the beta distribution with the density proportional to (1 − ‖ x ‖ 2) β , where x is belongs to the unit ball in R d ; (ii) the beta' distribution with the density proportional to (1 + ‖ x ‖ 2) − β on the whole of R d. These results imply explicit formulae for the expected face numbers of the following random polytopes: (a) the typical Poisson-Voronoi cell; (b) the zero cell of the Poisson hyperplane tessellation; (c) beta and beta' polytopes defined as convex hulls of i.i.d. samples from the corresponding distributions. [ABSTRACT FROM AUTHOR]