scholarly journals The normal distribution, an epistemological view

St open ◽  
2021 ◽  
Vol 2 ◽  
pp. 1-16
Author(s):  
Benjamin Benzon
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Distribution ◽  
Laws Of Nature ◽  
Mathematical Structure ◽  
Model Parameters ◽  
Close Relationship ◽  
Estimation Of Model Parameters ◽  
Von Mises

The role of the normal distribution in the realm of statistical inference and science is considered from epistemological viewpoint. Quantifiable knowledge is usually embodied in mathematical models. History and emergence of the normal distribution is presented in a close relationship to those models. Furthermore, the role of the normal distribution in estimation of model parameters, starting with Laplace’s Central Limit Theorem, through maximum likelihood theory leading to Bronstein von Mises and Convolution Theorems, is discussed. The paper concludes with the claim that our knowledge on the effects of variables in models or laws of nature has a mathematical structure which is identical to the normal distribution. The epistemological consequences of the latter claim are also considered.

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.

scholarly journals Central limit theorem for a class of random measures associated with germ-grain models

1996 ◽  
Vol 28 (02) ◽  
pp. 333-334
Author(s):  
Lothar Heinrich ◽  
Ilya S. Molchanov
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Fields ◽  
Boolean Model ◽  
Random Sets ◽  
Model Parameters ◽  
Random Measures ◽  
Intrinsic Volumes ◽  
Individual Grains

We introduce a family of stationary random measures in the Euclidean space generated by so-called germ-grain models. The germ-grain model is defined as the union of i.i.d. compact random sets (grains) shifted by points (germs) of a point process. This model gives rise to random measures defined by the sum of contributions of non-overlapping parts of the individual grains. The corresponding moment measures are calculated and an ergodic theorem is presented. The main result is the central limit theorem for the introduced random measures, which is valid for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. The technique is based on a central limit theorem for β-mixing random fields. It is shown that this construction of random measures includes those random measures obtained by the so-called positive extensions of intrinsic volumes. In the Poisson case it is possible to prove a central limit theorem under weaker assumptions by using approximations by m-dependent random fields. Applications to statistics of the Boolean model are also discussed. They include a standard way to derive limit theorems for estimators of the model parameters.

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.

scholarly journals The Role of the Central Limit Theorem in the Heterogeneous Ensemble of Brownian Particles Approach

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1145 ◽  
Author(s):  
Silvia Vitali ◽  
Iva Budimir ◽  
Claudio Runfola ◽  
Gastone Castellani
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Stable Distributions ◽  
The Central Limit Theorem ◽  
Brownian Particles ◽  
Heterogeneous Ensemble

The central limit theorem (CLT) and its generalization to stable distributions have been widely described in literature. However, many variations of the theorem have been defined and often their applicability in practical situations is not straightforward. In particular, the applicability of the CLT is essential for a derivation of heterogeneous ensemble of Brownian particles (HEBP). Here, we analyze the role of the CLT within the HEBP approach in more detail and derive the conditions under which the existing theorems are valid.

Central limit theorem for germination-growth models in ℝd with non-Poisson locations

2001 ◽  
Vol 33 (04) ◽  
pp. 751-755
Author(s):  
S. N. Chiu ◽  
M. P. Quine
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Distribution ◽  
Central Limit ◽  
Growth Models ◽  
Germination Time ◽  
Asymptotic Normal Distribution ◽  
Asymptotic Normal ◽  
Dependent Point ◽  
Germinated Seeds

Seeds are randomly scattered in ℝ d according to an m-dependent point process. Each seed has its own potential germination time. From each seed that succeeds in germinating, a spherical inhibited region grows to prohibit germination of any seed with later potential germination time. We show that under certain conditions on the distribution of the potential germination time, the number of germinated seeds in a large region has an asymptotic normal distribution.

scholarly journals Central limit theorem for a class of random measures associated with germ-grain models

1999 ◽  
Vol 31 (02) ◽  
pp. 283-314 ◽  
Author(s):  
Lothar Heinrich ◽  
Ilya S. Molchanov
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Boolean Model ◽  
Random Sets ◽  
Model Parameters ◽  
Random Measures ◽  
Intrinsic Volumes ◽  
The Individual ◽  
Individual Grains

The germ-grain model is defined as the union of independent identically distributed compact random sets (grains) shifted by points (germs) of a point process. The paper introduces a family of stationary random measures in ℝ d generated by germ-grain models and defined by the sum of contributions of non-overlapping parts of the individual grains. The main result of the paper is the central limit theorem for these random measures, which holds for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. It is shown that this construction of random measures includes those random measures obtained by positively extended intrinsic volumes. In the Poisson case it is possible to prove a central limit theorem under weaker assumptions by using approximations by m-dependent random fields. Applications to statistics of the Boolean model are also discussed. They include a standard way to derive limit theorems for estimators of the model parameters.

Central limit theorem for germination-growth models in ℝd with non-Poisson locations

2001 ◽  
Vol 33 (4) ◽  
pp. 751-755 ◽  
Author(s):  
S. N. Chiu ◽  
M. P. Quine
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Distribution ◽  
Central Limit ◽  
Growth Models ◽  
Germination Time ◽  
Asymptotic Normal Distribution ◽  
Asymptotic Normal ◽  
Dependent Point ◽  
Germinated Seeds

Seeds are randomly scattered in ℝd according to an m-dependent point process. Each seed has its own potential germination time. From each seed that succeeds in germinating, a spherical inhibited region grows to prohibit germination of any seed with later potential germination time. We show that under certain conditions on the distribution of the potential germination time, the number of germinated seeds in a large region has an asymptotic normal distribution.

On the role of the central limit theorem in 3 D quantum gravity

2017 ◽  
Vol 79 (2) ◽  
pp. 165-186
Author(s):  
Maria Simonetta Bernabei ◽  
Horst Thaler
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Quantum Gravity ◽  
Central Limit ◽  
The Central Limit Theorem

scholarly journals A Central Limit Theorem for M-estimators by the von Mises Method

1992 ◽  
Vol 46 (2-3) ◽  
pp. 165-177 ◽  
Author(s):  
C.C. Heesterman ◽  
R.D. Gill
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Von Mises
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