scholarly journals Central Limit Theorems for Volume and Surface Content of Stationary Poisson Cylinder Processes in Expanding Domains

2013 ◽  
Vol 45 (2) ◽  
pp. 312-331 ◽  
Author(s):  
Lothar Heinrich ◽  
Malte Spiess
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Euclidean Space ◽  
Asymptotic Behaviour ◽  
Central Limit ◽  
Limit Theorems ◽  
Dimensional Euclidean Space ◽  
Surface Content ◽  
Directional Distributions ◽  
Shaped Domain

A stationary Poisson cylinder process in the d-dimensional Euclidean space is composed of a stationary Poisson process of k-flats (0 ≤ k ≤ d−1) which are dilated by independent and identically distributed 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 ϱ W as ϱ grows unboundedly. Due to the long-range dependencies within the union set of cylinders, the variance of its d-volume in ϱ W increases asymptotically proportional to the (d+k) th power of ϱ. To obtain the exact asymptotic behaviour of this variance, we need a distinction between discrete and continuous directional distributions of the typical k-flat. A corresponding central limit theorem for the surface content is stated at the end.

scholarly journals Central Limit Theorems for Volume and Surface Content of Stationary Poisson Cylinder Processes in Expanding Domains

2013 ◽  
Vol 45 (02) ◽  
pp. 312-331
Author(s):  
Lothar Heinrich ◽  
Malte Spiess
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Euclidean Space ◽  
Asymptotic Behaviour ◽  
Central Limit ◽  
Limit Theorems ◽  
Dimensional Euclidean Space ◽  
Surface Content ◽  
Directional Distributions ◽  
Shaped Domain

A stationary Poisson cylinder process in the d-dimensional Euclidean space is composed of a stationary Poisson process of k-flats (0 ≤ k ≤ d−1) which are dilated by independent and identically distributed 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 ϱ W as ϱ grows unboundedly. Due to the long-range dependencies within the union set of cylinders, the variance of its d-volume in ϱ W increases asymptotically proportional to the (d+k) th power of ϱ. To obtain the exact asymptotic behaviour of this variance, we need a distinction between discrete and continuous directional distributions of the typical k-flat. A corresponding central limit theorem for the surface content is stated at the end.

scholarly journals Limit theorems for multivariate Brownian semistationary processes and feasible results

2019 ◽  
Vol 51 (03) ◽  
pp. 667-716
Author(s):  
Riccardo Passeggeri ◽  
Almut E. D. Veraart
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Behaviour ◽  
Central Limit ◽  
Gaussian Processes ◽  
Limit Theorems ◽  
Laws Of Large Numbers ◽  
Stationary Increments ◽  
Large Numbers ◽  
Weak Laws

AbstractIn this paper we introduce the multivariate Brownian semistationary (BSS) process and study the joint asymptotic behaviour of its realised covariation using in-fill asymptotics. First, we present a central limit theorem for general multivariate Gaussian processes with stationary increments, which are not necessarily semimartingales. Then, we show weak laws of large numbers, central limit theorems, and feasible results for BSS processes. An explicit example based on the so-called gamma kernels is also provided.

A central limit theorem by remainder term for renewal processes

1992 ◽  
Vol 24 (2) ◽  
pp. 267-287 ◽  
Author(s):  
Allen L. Roginsky
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Remainder Term ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Renewal Processes

Three different definitions of the renewal processes are considered. For each of them, a central limit theorem with a remainder term is proved. The random variables that form the renewal processes are independent but not necessarily identically distributed and do not have to be positive. The results obtained in this paper improve and extend the central limit theorems obtained by Ahmad (1981) and Niculescu and Omey (1985).

Heavy traffic limit theorems in fluctuation theory

1978 ◽  
Vol 10 (04) ◽  
pp. 852-866
Author(s):  
A. J. Stam
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
The Central Limit Theorem ◽  
Heavy Traffic Limit ◽  
Oscillating Random Walk

Let be a family of random walks with For ε↓0 under certain conditions the random walk U (∊) n converges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. Let M ∊ = max {U (∊) n , n ≧ 0}, v 0 = min {n : U (∊) n = M ∊}, v 1 = max {n : U (∊) n = M ∊}. The joint limiting distribution of ∊2σ∊ –2 v 0 and ∊σ∊ –2 M ∊ is determined. It is the same as for ∊2σ∊ –2 v 1 and ∊σ–2 ∊ M ∊. The marginal ∊σ–2 ∊ M ∊ gives Kingman's heavy traffic theorem. Also lim ∊–1 P(M ∊ = 0) and lim ∊–1 P(M ∊ < x) are determined. Proofs are by direct comparison of corresponding probabilities for U (∊) n and for a special family of random walks related to MI/M/1 queues, using the central limit theorem.

The convex hull of a normal sample

1994 ◽  
Vol 26 (04) ◽  
pp. 855-875 ◽  
Author(s):  
Irene Hueter
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Convex Hull ◽  
Asymptotic Behaviour ◽  
Central Limit ◽  
Random Variables ◽  
Poisson Approximation ◽  
Normal Sample ◽  
Central Importance ◽  
Independent Identically Distributed

Consider the convex hull of n independent, identically distributed points in the plane. Functionals of interest are the number of vertices Nn , the perimeter Ln and the area An of the convex hull. We study the asymptotic behaviour of these three quantities when the points are standard normally distributed. In particular, we derive the variances of Nn, Ln and An for large n and prove a central limit theorem for each of these random variables. We enlarge on a method developed by Groeneboom (1988) for uniformly distributed points supported on a bounded planar region. The process of vertices of the convex hull is of central importance. Poisson approximation and martingale techniques are used.

On the behaviour of a long cascade of linear reservoirs

2000 ◽  
Vol 37 (02) ◽  
pp. 417-428
Author(s):  
John E. Glynn ◽  
Peter W. Glynn
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Large Deviations ◽  
Asymptotic Behaviour ◽  
Central Limit ◽  
Tail Probabilities ◽  
Logarithmic Asymptotics ◽  
Random Fluctuations

This paper describes the limiting asymptotic behaviour of a long cascade of linear reservoirs fed by stationary inflows into the first reservoir. We show that the storage in the nth reservoir becomes asymptotically deterministic as n → ∞, and establish a central limit theorem for the random fluctuations about the deterministic approximation. In addition, we prove a large deviations theorem that provides precise logarithmic asymptotics for the tail probabilities associated with the storage in the nth reservoir when n is large.

scholarly journals Central limit theorems for functionals of stationary germ-grain models

2006 ◽  
Vol 38 (1) ◽  
pp. 76-94 ◽  
Author(s):  
Ursa Pantle ◽  
Volker Schmidt ◽  
Evgueni Spodarev
Keyword(s):  
Limit Theorem ◽  
Central Limit ◽  
Random Fields ◽  
Limit Theorems ◽  
Boolean Model ◽  
Dimensional Euclidean Space ◽  
Boolean Models ◽  
Intrinsic Volumes ◽  
Integrability Conditions ◽  
Vector Valued

Conditions are derived for the asymptotic normality of a general class of vector-valued functionals of stationary Boolean models in the d-dimensional Euclidean space, where a Lindeberg-type central limit theorem for m-dependent random fields, m ∈ N, is applied. These functionals can be used to construct joint estimators for the vector of specific intrinsic volumes of the underlying Boolean model. Extensions to functionals of more general germ–grain models satisfying some mixing and integrability conditions are also discussed.

scholarly journals A central limit theorem for numbers satisfying a class of triangular arrays associated with Hermite polynomials

2021 ◽  
Vol 61 ◽  
pp. 1-7
Author(s):  
Igoris Belovas
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Convergence Rate ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Generating Function ◽  
Limit Theorems ◽  
Hermite Polynomials ◽  
Triangular Arrays ◽  
Analytical Expressions

The paper extends the investigations of limit theorems for numbers satisfying a class of triangular arrays. We obtain analytical expressions for the semiexponential generating function the numbers, associated with Hermite polynomials. We apply the results to prove the asymptotic normality of the numbers and specify the convergence rate to the limiting distribution.    

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