Schreiber and Yukich [Ann. Probab. 36 (2008) establish an asymptotic representation for random convex polytope geometry in the unit ball B d , d ≥ 2, in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of generalized paraboloid growth processes. This paper further exploits this connection, introducing also a dual object termed the paraboloid hull process. Via these growth processes we establish local functional limit theorems for the properly scaled radius-vector and support functions of convex polytopes generated by high-density Poisson samples. We show that direct methods lead to explicit asymptotic expressions for the fidis of the properly scaled radius-vector and support functions. Generalized paraboloid growth processes, coupled with general techniques of stabilization theory, yield Brownian sheet limits for the defect volume and mean width functionals. Finally we provide explicit variance asymptotics and central limit theorems for the k-face and intrinsic volume functionals.
Given a Gibbs point process P Ψ on R d having a weak enough potential Ψ, we consider the random measures µ λ := P x∈P Ψ ∩Q λ ξ(x, P Ψ ∩ Q λ )δ x/λ 1/d , where Q λ := [−λ 1/d /2, λ 1/d /2] d is the volume λ cube and where ξ(·, ·) is a translation invariant stabilizing functional. Subject to Ψ satisfying a localization property and translation invariance, we establish weak laws of large numbers for λ −1 µ λ (f ), f a bounded test function on R d , and weak convergence of λ −1/2 µ λ (f ), suitably centered, to a Gaussian field acting on bounded test functions. The result yields limit laws for geometric functionals on Gibbs point processes including the Strauss and area interaction point processes as well as more general point processes defined by the Widom-Rowlinson and hard-core model. We provide applications to random sequential packing on Gibbsian input, to functionals of Euclidean graphs, networks, and percolation models on Gibbsian input, and to quantization via Gibbsian input. American Mathematical Society 2000 subject classifications. Primary 60F05, 60G55, Secondary 60D051 where X ⊂ R d is locally finite and where the function ξ, defined on all pairs (x, X ), with x ∈ X , represents the interaction of x with respect to X . When X is a random n point set in R d (i.e. a finite spatial point process), the asymptotic analysis of the suitably scaled sums (1.1) as n → ∞ can often be handled by M -dependent methods, ergodic theory, or mixing methods. However there are situations where these classical methods are either not directly applicable, do not give explicit asymptotics in terms of underlying geometry and point densities, or do not easily yield explicit rates of convergence. Stabilization methods originating in [23] and further developed in [3,24,26], provide another approach for handling sums of spatially dependent terms.There are several similar definitions of stabilization, but the essence is captured by the notion of stabilization of the functional ξ with respect to a rate τ > 0 homogeneous Poisson point processfor all z ∈ R d . Let B r (x) denote the Euclidean ball centered at x with radius r ∈ R + := [0, ∞).Letting 0 denote the origin of R d , we say that a translation invariant ξ is stabilizing on P = P τ if there exists an a.s. finite random variable R := R ξ (τ ) (a 'radius of stabilization') such thatConsider the point measures3) where δ x denotes the unit Dirac point mass at x whereas Q λ := [−λ 1/d /2, λ 1/d /2] d is the λ-volume cube. Let B(Q 1 ) denote the class of all bounded f : Q 1 → R and for all random point measures µ on R d let f, µ := f dµ and letμ := µ − E [µ]. Stabilization of translation invariant ξ on P, as defined in (1.2), together with stabilization of ξ on P ∩ Q λ , λ ≥ 1, when combined with appropriate moment conditions on ξ, yields for all f ∈ B(Q 1 ) the law of large numbers [22, 25] lim λ→∞ λ −1 f, µ λ = τ E [ξ(0, P)] Q1 f (x)dx in L 1 and in L 2 , (1.4) and, if the stabilization radii on P and P ∩ Q λ , λ ≥ 1, decay exponentially fast, then [3, 21] lim λ→∞ λ −1 Var[ f, µ λ ] ...
We show that the random point measures induced by vertices in the convex hull of a Poisson sample on the unit ball, when properly scaled and centered, converge to those of a mean zero Gaussian field. We establish limiting variance and covariance asymptotics in terms of the density of the Poisson sample. Similar results hold for the point measures induced by the maximal points in a Poisson sample. The approach involves introducing a generalized spatial birth growth process allowing for cell overlap.
The purpose of the present paper is to establish explicit bounds on moderate deviation probabilities for a rather general class of geometric functionals enjoying the stabilization property, under Poisson input and the assumption of a certain control over the growth of the moments of the functional and its radius of stabilization. Our proof techniques rely on cumulant expansions and cluster measures and yield completely explicit bounds on deviation probabilities. In addition, we establish a new criterion for the limiting variance to be non-degenerate. Moreover, our main result provides a new central limit theorem, which, though stated under strong moment assumptions, does not require bounded support of the intensity of the Poisson input. We apply our results to three groups of examples: random packing models, geometric functionals based on Euclidean nearest neighbors and the sphere of influence graphs.Comment: 52 page
We construct random dynamics for collections of nonintersecting planar contours, leaving invariant the distributions of length- and area-interacting polygonal Markov fields with V-shaped nodes. The first of these dynamics is based on the dynamic construction of consistent polygonal fields, as presented in the original articles by Arak (1983) and Arak and Surgailis (1989), (1991), and it provides an easy-to-implement Metropolis-type simulation algorithm. The second dynamics leads to a graphical construction in the spirit of Fernández et al. (1998), (2002) and yields a perfect simulation scheme in a finite window in the infinite-volume limit. This algorithm seems difficult to implement, yet its value lies in that it allows for theoretical analysis of the thermodynamic limit behaviour of length-interacting polygonal fields. The results thus obtained include, in the class of infinite-volume Gibbs measures without infinite contours, the uniqueness and exponential α-mixing of the thermodynamic limit of such fields in the low-temperature region. Outside this class, we conjecture the existence of an infinite number of extreme phases breaking both the translational and rotational symmetries.
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