scholarly journals Limit theorems for generalized perimeters of random inscribed polygons. II

2021 ◽  
Vol 8 (1) ◽  
pp. 101-110
Author(s):  
Ekaterina N. Simarova ◽  
◽  
Keyword(s):  
Stochastic Geometry ◽  
Scalar Product ◽  
Limit Theorems ◽  
Extreme Values ◽  
Limit Behavior ◽  
Maximum Diameter ◽  
Applied Method ◽  
Generalized Perimeter ◽  
Definition Of ◽  
Inscribed Polygons

Lao and Mayer (2008) recently developed the theory of U-max statistics, where instead of the usual sums over subsets, the maximum of the kernel is considered. Such statistics often appear in stochastic geometry. Examples include the greatest distance between random points in a ball, the maximum diameter of a random polygon, the largest scalar product in a sample of points, etc. Their limit distributions are related to distribution of extreme values. This is the second article devoted to the study of the generalized perimeter of a polygon and the limit behavior of the U-max statistics associated with the generalized perimeter. Here we consider the case when the parameter y, arising in the definition of the generalized perimeter, is greater than 1. The problems that arise in the applied method in this case are described. The results of theorems on limit behavior in the case of a triangle are refined.

scholarly journals Limit theorems for generalized perimeters of random inscribed polygons. I

2020 ◽  
Vol 65 (4) ◽  
pp. 678-687
Author(s):  
Ekaterina N. Simarova ◽  
◽  
Keyword(s):  
Stochastic Geometry ◽  
Limit Theorems ◽  
Extreme Values ◽  
Limiting Behavior ◽  
Limit Distributions ◽  
Random Polygon ◽  
Generalized Perimeter ◽  
Definition Of ◽  
Inscribed Polygons

Lao and Mayer (2008) recently developed the theory of U-max-statistics, where instead of the usual averaging the values of the kernel over subsets, the maximum of the kernel is considered. Such statistics often appear in stochastic geometry. Their limit distributions are related to distributions of extreme values. This is the first article devoted to the study of the generalized perimeter (the sum of side powers) of an inscribed random polygon, and of U-max-statistics associated with it. It describes the limiting behavior for the extreme values of the generalized perimeter. This problem has not been studied in the literature so far. One obtains some limit theorems in the case when the parameter y, arising in the definition of the generalized perimeter does not exceed 1.

A stress sensitivity model for the permeability of porous media based on bi-dispersed fractal theory

2018 ◽  
Vol 29 (02) ◽  
pp. 1850019 ◽  
Author(s):  
X.-H. Tan ◽  
C.-Y. Liu ◽  
X.-P. Li ◽  
H.-Q. Wang ◽  
H. Deng
Keyword(s):  
Porous Media ◽  
Fractal Dimension ◽  
Fractal Theory ◽  
Stress Sensitivity ◽  
Maximum Diameter ◽  
Proposed Model ◽  
Fractal Parameters ◽  
Stress Changes ◽  
Capillary Bundle ◽  
Definition Of

A stress sensitivity model for the permeability of porous media based on bidispersed fractal theory is established, considering the change of the flow path, the fractal geometry approach and the mechanics of porous media. It is noted that the two fractal parameters of the porous media construction perform differently when the stress changes. The tortuosity fractal dimension of solid cluster [Formula: see text] become bigger with an increase of stress. However, the pore fractal dimension of solid cluster [Formula: see text] and capillary bundle [Formula: see text] remains the same with an increase of stress. The definition of normalized permeability is introduced for the analyzation of the impacts of stress sensitivity on permeability. The normalized permeability is related to solid cluster tortuosity dimension, pore fractal dimension, solid cluster maximum diameter, Young’s modulus and Poisson’s ratio. Every parameter has clear physical meaning without the use of empirical constants. Predictions of permeability of the model is accordant with the obtained experimental data. Thus, the proposed model can precisely depict the flow of fluid in porous media under stress.

scholarly journals Some recent advances for limit theorems

2020 ◽  
Vol 68 ◽  
pp. 73-96 ◽  
Author(s):  
Benjamin Arras ◽  
Jean-Christophe Breton ◽  
Aurelia Deshayes ◽  
Olivier Durieu ◽  
Raphaël Lachièze-Rey
Keyword(s):  
Probability Theory ◽  
Stochastic Geometry ◽  
Limit Theorems ◽  
Growth Models ◽  
Rates Of Convergence ◽  
Discrete Models ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Correlated Random Walks ◽  
Recent Developments

We present some recent developments for limit theorems in probability theory, illustrating the variety of this field of activity. The recent results we discuss range from Stein’s method, as well as for infinitely divisible distributions as applications of this method in stochastic geometry, to asymptotics for some discrete models. They deal with rates of convergence, functional convergences for correlated random walks and shape theorems for growth models.

SOME LIMIT THEOREMS OF DELAYED AVERAGES FOR COUNTABLE NONHOMOGENEOUS MARKOV CHAINS

2019 ◽  
pp. 1-19
Author(s):  
Pingping Zhong ◽  
Weiguo Yang ◽  
Zhiyan Shi ◽  
Yan Zhang
Keyword(s):  
Information Theory ◽  
Markov Chains ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Strong Ergodicity ◽  
Large Numbers ◽  
Nonhomogeneous Markov Chains ◽  
Definition Of ◽  
Bivariate Functions

AbstractThe purpose of this paper is to establish some limit theorems of delayed averages for countable nonhomogeneous Markov chains. The definition of the generalized C-strong ergodicity and the generalized uniformly C-strong ergodicity for countable nonhomogeneous Markov chains is introduced first. Then a theorem about the generalized C-strong ergodicity and the generalized uniformly C-strong ergodicity for the nonhomogeneous Markov chains is established, and its applications to the information theory are given. Finally, the strong law of large numbers of delayed averages of bivariate functions for countable nonhomogeneous Markov chains is proved.

scholarly journals DIFFUSION LIMITS FOR A MARKOV MODULATED BINOMIAL COUNTING PROCESS

2019 ◽  
Vol 34 (2) ◽  
pp. 235-257
Author(s):  
Peter Spreij ◽  
Jaap Storm
Keyword(s):  
Markov Chain ◽  
Markov Process ◽  
Regime Switching ◽  
Limit Theorems ◽  
Counting Process ◽  
Limit Behavior ◽  
Diffusion Approximations ◽  
Default Rate ◽  
Markov Modulated ◽  
Rapid Switching

In this paper, we study limit behavior for a Markov-modulated binomial counting process, also called a binomial counting process under regime switching. Such a process naturally appears in the context of credit risk when multiple obligors are present. Markov-modulation takes place when the failure/default rate of each individual obligor depends on an underlying Markov chain. The limit behavior under consideration occurs when the number of obligors increases unboundedly, and/or by accelerating the modulating Markov process, called rapid switching. We establish diffusion approximations, obtained by application of (semi)martingale central limit theorems. Depending on the specific circumstances, different approximations are found.

scholarly journals THE SOq(N, R)-SYMMETRIC HARMONIC OSCILLATOR ON THE QUANTUM EUCLIDEAN SPACE ${\bf R}_q^N$ AND ITS HILBERT SPACE STRUCTURE

1993 ◽  
Vol 08 (26) ◽  
pp. 4679-4729 ◽  
Author(s):  
GAETANO FIORE
Keyword(s):  
Hilbert Space ◽  
Euclidean Space ◽  
Harmonic Oscillator ◽  
Mechanical Model ◽  
Scalar Product ◽  
Space Structure ◽  
Quantum Mechanical ◽  
Quantum Space ◽  
Isotropic Harmonic Oscillator ◽  
Definition Of

We show that the isotropic harmonic oscillator in the ordinary Euclidean space RN (N≥3) admits a natural q-deformation into a new quantum-mechanical model having a q-deformed symmetry (in the sense of quantum groups), SO q(N, R). The q-deformation is the consequence of replacing RN by [Formula: see text] (the corresponding quantum space). This provides an example of quantum mechanics on a noncommutative geometrical space. To reach the goal, we also have to deal with a sensible definition of integration over [Formula: see text], which we use for the definition of the scalar product of states.

scholarly journals Limit Theorems for the Inductive Mean on Metric Trees

2010 ◽  
Vol 47 (04) ◽  
pp. 1136-1149 ◽  
Author(s):  
Bojan Basrak
Keyword(s):  
Strong Consistency ◽  
Limit Theorems ◽  
Random Variables ◽  
Expected Value ◽  
Asymptotic Distributions ◽  
Weighted Interpolation ◽  
Underlying Distribution ◽  
Data Point ◽  
Definition Of

For random variables with values on binary metric trees, the definition of the expected value can be generalized to the notion of a barycenter. To estimate the barycenter from tree-valued data, the so-called inductive mean is constructed recursively using the weighted interpolation between the current mean and a new data point. We show the strong consistency of the inductive mean, but also that it, somewhat peculiarly, converges towards the true barycenter with different rates, and asymptotic distributions depending on the small variations of the underlying distribution.

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