Variance asymptotics for the area of planar cylinder processes generated by Brillinger-mixing point processes (original) (raw)
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On the Variance of the Area of Planar Cylinder Processes Driven by Brillinger-Mixing Point Processes
2021
We study some asymptotic properties of cylinder processes in the plane defined as union sets of dilated straight lines (appearing as mutually overlapping infinitely long strips) derived from a stationary independently marked point process on the real line, where the marks describe thickness and orientation of individual cylinders. Such cylinder processes form an important class of (in general non-stationary) planar random sets. We observe the cylinder process in an unboundedly growing domain ̺K when ̺ → ∞ , where the set K is compact and star-shaped w.r.t. the origin o being an inner point of K. Provided the unmarked point process satisfies a Brillinger-type mixing condition and the thickness of the typical cylinder has a finite second moment we prove a (weak) law of large numbers as well as a formula of the asymptotic variance for the area of the cylinder process in ̺K. Due to the long-range dependencies of the cylinder process, this variance increases proportionally to ̺.
Absolute regularity and Brillinger-mixing of stationary point processes
Lithuanian Mathematical Journal, 2013
We study the following problem: How to verify Brillinger-mixing of stationary point processes in R d by imposing conditions on a suitable mixing coefficient? For this, we define an absolute regularity (or β-mixing) coefficient for point processes and derive, in terms of this coefficient, an explicit condition that implies finite total variation of the kth-order reduced factorial cumulant measure of the point process for fixed k 2. To prove this, we introduce higher-order covariance measures and use Statulevičius' representation formula for mixed cumulants in case of random (counting) measures. To illustrate our results, we consider some Brillinger-mixing point processes occurring in stochastic geometry.
Brillinger-mixing point processes need not to be ergodic
Statistics & Probability Letters, 2018
Mixing and ergodicity Poisson cluster processes Moment problem for point processes a b s t r a c t Recently, it has been proved that a stationary Brillinger-mixing point process is mixing (of any order) if its moment measures determine the distribution uniquely. In this paper we construct a family of non-ergodic stationary point processes as mixture of two distinct Brillinger-mixing Neyman-Scott processes having the same moment measures.
Central Limit Theorem for the Volume of stationary Poisson Cylinder Processes in Expanding Domains
A stationary Poisson cylinder process in the d-dimensional Euclidean space is composed by a stationary Poisson process of k-flats (1 ≤ k ≤ d − 1) which are dilated by i.i.d. random compact cylinder bases taken from the corresponding (d − k)-dimensional orthogonal complement. If the second moment of the (d − k)-volume of the typical cylinder base exists, we prove asymptotic normality of the d-volume of the union set of Poisson cylinders that covers an expanding star-shaped domain ̺ W as ̺ grows unboundedly. Due to the long-range dependences within the union set of cylinders, the variance of its d-volume in ̺ W increases asymptotically proportional to the (d + k)th power of ̺. To obtain the exact asymptotic behaviour of this variance we need a distinction between discrete and continuous directional distributions of the typical k-flat.
On an extension of Lévy's stochastic area process to higher dimensions
Stochastic Processes and their Applications, 1993
For n 2 2 an (n -I)-parameter real process V,,, called stochastic volume, is defined. This process is an extension to higher dimensions of L&y's stochastic area which is obtained from V,, by setting n = 2. For V,, a Strassen-type functional law of the iterated logarithm is proved by making use of large deviations techniques.
Lithuanian Mathematical Journal, 2009
A stationary Poisson cylinder process Π (d,k) cyl is composed of a stationary Poisson process of k-flats in R d that are dilated by i.i.d. random compact cylinder bases taken from the corresponding orthogonal complement. We study the accuracy of normal approximation of the d-volume V (d,k) of the union set of Π (d,k) cyl that covers W as the scaling factor becomes large. Here W is some fixed compact star-shaped set containing the origin as an inner point. We give lower and upper bounds of the variance of V (d,k) that exhibit long-range dependence within the union set of cylinders. Our main results are sharp estimates of the higher-order cumulants of V (d,k) under the assumption that the (d − k)-volume of the typical cylinder base possesses a finite exponential moment. These estimates enable us to apply the celebrated "Lemma on large deviations" of Statulevičius.
Fluctuations of shapes of large areas under paths of random walks
Probability Theory and Related Fields, 1996
We discuss statistical properties of random walks conditioned by fixing a large area under their paths. We prove the functional central limit theorem (invariance principle) for these conditional distributions. The limiting Gaussian measure coincides with the conditional probability distribution of certain time-nonhomogeneous Gaussian random process obtained by an integral transformation of the white noise. From the point of view of statistical mechanics the studied problem is the problem of describing the fluctuations of the phase boundary in the one-dimensional SOS-model.
Limit theory for geometric statistics of point processes having fast decay of correlations
The Annals of Probability
Let P be a simple, stationary point process on R d having fast decay of correlations, i.e., its correlation functions factorize up to an additive error decaying faster than any power of the separation distance. Let Pn := P ∩ Wn be its restriction to windows Wn := [− 1 2 n 1/d , 1 2 n 1/d ] d ⊂ R d. We consider the statistic H ξ n := x∈Pn ξ(x, Pn) where ξ(x, Pn) denotes a score function representing the interaction of x with respect to Pn. When ξ depends on local data in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics, and central limit theorems for H ξ n and, more generally, for statistics of the re-scaled, possibly signed, ξ-weighted point measures µ ξ n := x∈Pn ξ(x, Pn)δ n −1/d x , as Wn ↑ R d. This gives the limit theory for non-linear geometric statistics (such as clique counts, the number of Morse critical points, intrinsic volumes of the Boolean model, and total edge length of the k-nearest neighbors graph) of α-determinantal point processes (for −1/α ∈ N) having fast decreasing kernels, including the β-Ginibre ensembles, extending the Gaussian fluctuation results of Soshnikov [72] to non-linear statistics. It also gives the limit theory for geometric U-statistics of α-permanental point processes (for 1/α ∈ N) as well as the zero set of Gaussian entire functions, extending the central limit theorems of Nazarov and Sodin [53] and Shirai and Takahashi [71], which are also confined to linear statistics. The proof of the central limit theorem relies on a factorial moment expansion originating in [12, 13] to show the fast decay of the correlations of ξ-weighted point measures. The latter property is shown to imply a condition equivalent to Brillinger mixing and consequently yields the asymptotic normality of µ ξ n via an extension of the cumulant method. CONTENTS
Winding of planar Brownian curves
Journal of Physics A: Mathematical and General, 1990
We compute the joint probability for a closed Brownian curve to wind n times around a prescribed point and to enclose a given algebraic area. An estimate from below of the arithmetic area is obtained. Since the pioneering work of Edwards [l], the study of path integrals in the presence of topological constraints has aroused considerable interest. On the one hand these techniques are of direct relevance for polymer physics while on the other hand they are connected with some rigorous mathematical results. Consider for instance the two-dimensional Brownian motion on the punctured plane P-(0). The problem of finding the asymptotic probability distribution of the total angle e (t) wound at time t around 0 was first addressed by Spitzer [2] who showed that X =: 2e(t)/ln t is distributed according to a Cauchy law for t + +CO. This result was then extended by Pitman and Yor [3] to the case of n prescribed points. This question has also been reexamined by Rudnick and Hu [4] who showed that by removing a disc from the plane, instead of a point, the asymptotic distribution changes drastically from a Cauchy law to an exponential law (which thus leads to finite moments). Recent results of Belisle [ 5 ] on the winding of a discrete random walk indeed confirm that the limiting law has an exponential tail. The winding number distribution was also discussed by Wiegel in the context of polymer entanglements [ 6 ]. An apparently unrelated problem concerns the probability distribution of the area enclosed by a planar Brownian curve. First raised by Levy [7] and solved magisterially by the use of Fourier-Wiener series, this problem was more recently reexamined by Brereton and Butler [8], Khandekar and Wiegel [ 9 ] and Duplantier [lo]. The purpose of this paper is to extend this approach to the case of the joint probability distribution for a closed planar Brownian walk to wind n times around a prescribed point and enclose a given algebraic area (the initial = final point has been left unspecified). Interestingly enough, this quantity is related to the two-body partition function of a gas of particles obeying fractional statistics (anyons). The plan of the paper is as follows: for pedagogical reasons we first rederive Wiegel's results concerning the probability 9 (A) for a closed planar Brownian curve to enclose after a time 7 a given algebraic area A. This quantity can be expressed in terms of the partition function of a charged particle embedded in a constant magnetic field. This partition function diverges as the total area of the plane but an adequate normalisation leads back to the finite P (A). We then consider the probability B,(n)