scholarly journals “Normal” Orientation Distributions

1992 ◽  
Vol 19 (4) ◽  
pp. 197-202 ◽  
Author(s):  
H. Schaeben
Keyword(s):  
Probability Theory ◽  
Brownian Motion ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Euclidean Space ◽  
Normal Distribution ◽  
The Central Limit Theorem ◽  
Bingham Distribution ◽  
Normal Orientation ◽  
Orientation Distributions

Analogues of the normal distribution in Euclidean space for orientations represented by Rodrigues parameters are discussed. It is emphasized that different characterizations of the normal distribution in Euclidean space lead to different distributions in other spaces, none of which is mathematically superior to any other one. Particular analogues of the normal distribution are the Bingham distribution on S+4 for the purposes of mathematical statistics, and the Brownian motion distribution on S+4 in terms of probability theory and stochastic processes. It is reminded of the fact that a simple analogue of the central limit theorem in Euclidean space does not exist for the hyperspheres SP and projective hyperplanes HP−1=S+4.

scholarly journals On the Normal Distribution in the Orientation Space

1988 ◽  
Vol 10 (1) ◽  
pp. 77-96 ◽  
Author(s):  
S. Matthies ◽  
J. Muller ◽  
G. W. Vinel
Keyword(s):  
Probability Theory ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Distribution ◽  
Texture Analysis ◽  
Practical Interest ◽  
Orientation Space ◽  
The Central Limit Theorem ◽  
Very High ◽  
Approximative Expression

The properties of model distributions used in texture analysis up to now are discussed. The normal distribution in the G-space (recently investigated by T. I. Savjolova) is analysed. Its connection with the central limit theorem of probability theory is demonstrated in a mathematically simplified manner. An analytically closed approximative expression (with very high precision for halfwidths of practical interest) for the normal distribution is derived. Possible correlations between forms of texture components and mechanisms of texture development are mentioned.

scholarly journals Central and Local Limit Theorems for Numbers of the Tribonacci Triangle

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 880
Author(s):  
Igoris Belovas
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Distribution ◽  
Rate Of Convergence ◽  
Limit Theorems ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Constructive Proof ◽  
The Central Limit Theorem ◽  
Local Limit Theorems

In this research, we continue studying limit theorems for combinatorial numbers satisfying a class of triangular arrays. Using the general results of Hwang and Bender, we obtain a constructive proof of the central limit theorem, specifying the rate of convergence to the limiting (normal) distribution, as well as a new proof of the local limit theorem for the numbers of the tribonacci triangle.

The Patterns of Large Numbers

2019 ◽  
pp. 43-66
Author(s):  
Steven J. Osterlind
Keyword(s):  
Probability Theory ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
French Revolution ◽  
Central Limit ◽  
Historical Context ◽  
Binomial Theorem ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
First Time

This chapter advances the historical context for quantification by describing the climate of the day—social, cultural, political, and intellectual—as fraught with disquieting influences. Forces leading to the French Revolution were building, and the colonists in America were fighting for secession from England. During this time, three important number theorems came into existence: the binomial theorem, the law of large numbers, and the central limit theorem. Each is described in easy-to-understand language. These are fundamental to how numbers operate in a probability circumstance. Pascal’s triangle is explained as a shortcut solving some binomial expansions, and Jacob Bernoulli’s Ars Conjectandi, which presents the study of measurement “error” for the first time, is discussed. In addition, the central limit theorem is explained in terms of its relevance to probability theory, and its utility today.

scholarly journals The central limit theorem for $\sum f(\theta^nx)g(\theta^{n^2}x)$

2000 ◽  
Vol 20 (5) ◽  
pp. 1335-1353 ◽  
Author(s):  
KATUSI FUKUYAMA
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Distribution ◽  
Central Limit ◽  
The Central Limit Theorem ◽  
Distribution Of Values

In this paper, it is proved that the distribution of values of $N^{-1/2}\sum_{n=1}^N f_1(\theta^{p_1(n)}x)\dots f_K(\theta^{p_K(n)}x)$ converges to normal distribution. Here $p_k(n)$ are polynomials.

Moderate deviation for super-Brownian Motion with super-Brownian immigration

2002 ◽  
Vol 39 (04) ◽  
pp. 829-838 ◽  
Author(s):  
Wen-Ming Hong
Keyword(s):  
Brownian Motion ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Large Deviation ◽  
Moderate Deviation ◽  
The Central Limit Theorem ◽  
Large Deviation Principles ◽  
Super Brownian Motion ◽  
Random Immigration

Moderate deviation principles are established in dimensionsd≥ 3 for super-Brownian motion with random immigration, where the immigration rate is governed by the trajectory of another super-Brownian motion. It fills in the gap between the central limit theorem and large deviation principles for this model which were obtained by Hong and Li (1999) and Hong (2001).

scholarly journals ESTIMATION OF THE NUMBER OF SUMMANDS OF THE CENTRAL LIMIT THEOREM

2020 ◽  
pp. 7-19
Author(s):  
Antonina Ganicheva ◽  
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Distribution ◽  
Central Limit ◽  
Random Variables ◽  
Independent Random Variables ◽  
Distribution Law ◽  
The Central Limit Theorem ◽  
Sample Average ◽  
Normal Probability Distribution

The problem of estimating the number of summands of random variables for a total normal distribution law or a sample average with a normal distribution is investigated. The Central limit theorem allows us to solve many complex applied problems using the developed mathematical apparatus of the normal probability distribution. Otherwise, we would have to operate with convolutions of distributions that are explicitly calculated in rare cases. The purpose of this paper is to theoretically estimate the number of terms of the Central limit theorem necessary for the sum or sample average to have a normal probability distribution law. The article proves two theorems and two consequences of them. The method of characteristic functions is used to prove theorems. The first theorem States the conditions under which the average sample of independent terms will have a normal distribution law with a given accuracy. The corollary of the first theorem determines the normal distribution for the sum of independent random variables under the conditions of theorem 1. The second theorem defines the normal distribution conditions for the average sample of independent random variables whose mathematical expectations fall in the same interval, and whose variances also fall in the same interval. The corollary of the second theorem determines the normal distribution for the sum of independent random variables under the conditions of theorem 2. According to the formula relations proved in theorem 1, a table of the required number of terms in the Central limit theorem is calculated to ensure the specified accuracy of approximation of the distribution of the values of the sample average to the normal distribution law. A graph of this dependence is constructed. The dependence is well approximated by a polynomial of the sixth degree. The relations and proved theorems obtained in the article are simple, from the point of view of calculations, and allow controlling the testing process for evaluating students ' knowledge. They make it possible to determine the number of experts when making collective decisions in the economy and organizational management systems, to conduct optimal selective quality control of products, to carry out the necessary number of observations and reasonable diagnostics in medicine.

scholarly journals Central limit theorem for a class of SPDEs

2015 ◽  
Vol 52 (3) ◽  
pp. 786-796 ◽  
Author(s):  
Parisa Fatheddin
Keyword(s):  
Brownian Motion ◽  
Central Limit Theorem ◽  
Partial Differential Equations ◽  
Limit Theorem ◽  
Differential Equations ◽  
Central Limit ◽  
Stochastic Partial Differential Equations ◽  
Population Models ◽  
The Central Limit Theorem ◽  
Super Brownian Motion

In this paper we establish the central limit theorem for a class of stochastic partial differential equations and as an application derive this theorem for two widely studied population models: super-Brownian motion and the Fleming-Viot process.

Probability for Everyone—Even Philosophers

2017 ◽  
Author(s):  
Alan Hájek ◽  
Christopher Hitchcock
Keyword(s):  
Probability Theory ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Conditional Probability ◽  
Central Limit ◽  
Measure Theory ◽  
Bayes Theorem ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Key Concepts

In this chapter the basics of probability theory are introduced, with particular attention to those topics that are most important for applications in philosophy. The formalism is described in two passes. The first presents finite probability, which suffices for most philosophical discussions of probability. The second presents measure theory, which is needed for applications involving infinities or limits. Key concepts such as conditional probability, probabilistic independence, random variables, and expectation are defined. In addition, several important theorems, including Bayes’ theorem, the weak and strong laws of large numbers, and the central limit theorem are defined. Along the way, several familiar puzzles or paradoxes involving probability are discussed.

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