Consider a stationary Poisson process \(\eta\) in the d-dimensional Euclidean or hyperbolic space and construct a random graph with vertex set \(\eta\) as follows. First, each point \(x\in\eta\) is connected by an edge to its nearest neighbour, then to its second nearest neighbour and so on, until \(x\) is contained in the convex hull of the points already connected to . The resulting random graph is the so-called nearest neighbour embracing graph. The main result of this paper is a quantitative description of the Gaussian fluctuations of geometric functionals associated with the nearest neighbour embracing graph. More precisely, the total edge length, more general length-power functionals and the number of vertices with given outdegree are considered.

This paper deals with sequences of random variables \(X_n\) only taking values in \(\{0,\ldots,n\}\). The probability generating functions of such random variables are polynomials of degree \(n\). Under the assumption that the roots of these polynomials are either all real or all lie on the unit circle in the complex plane, a quantitative normal approximation bound for \(X_n\) is established in a unified way. In the real rooted case the result is classical and only involves the variances of \(X_n\), while in the cyclotomic case the fourth cumulants or moments of \(X_n\) appear in addition. The proofs are elementary and based on the Stein--Tikhomirov method.

An arrangement of hypersurfaces in projective space is SNC if and only if its Euler discriminant is nonzero. We study the critical loci of all Laurent monomials in the equations of the smooth hypersurfaces. These loci form an irreducible variety in the product of two projective spaces, known in algebraic statistics as the likelihood correspondence and in particle physics as the scattering correspondence. We establish an explicit determinantal representation for the bihomogeneous prime ideal of this variety.

For the distributions of finitely many binary random variables, we study the interaction of restrictions of the supports with conditional independence constraints. We prove a generalization of the Hammersley-Clifford theorem for distributions whose support is a natural distributive lattice: that is, any distribution which has natural lattice support and satisfies the pairwise Markov statements of a graph must factor according to the graph. We also show a connection to the Hibi ideals of lattices. dui fames leo sodales risus, posuere justo curae nostra, luctus vitae montes potenti, lectus sem.