On the behaviour of a long cascade of linear reservoirs

2000 ◽  
Vol 37 (2) ◽  
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.

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.

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.

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.

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.

The convex hull of a normal sample

1994 ◽  
Vol 26 (4) ◽  
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.

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