scholarly journals Optimal averaging procedures in almost sure central limit theory

2005 ◽  
Vol 2 (2) ◽  
Author(s):  
Siegfried Hörmann
Keyword(s):  
Distribution Function ◽  
Central Limit ◽  
Complete Solution ◽  
Standard Normal Distribution ◽  
Numerical Sequence ◽  
Normal Distribution Function ◽  
Standard Normal Distribution Function ◽  
Limit Theory ◽  
Logarithmic Means ◽  
Standard Normal

Let \(X_1, X_2, \ldots\) be i.i.d. random variables with \(\mathbb{E}[X_1] = 0\), \(\mathbb{E}[X_1^2] = 1\), \(S_n = X_1 + \cdots + X_n\) and let \((d_k)\) be a positive numerical sequence. We investigate the a.s. convergence of the averages \[\frac{1}{D_N} \sum_{k = 1}^{N} d_k I \{S_k / \sqrt{k} \leq x\},\]where \(D_N = \sum_{k = 1}^{N} d_k\). In the case of \(d_k = 1/k\) we have logarithmic means and by the almost sure central limit theorem the above averages converge a.s. to \(\Phi(x)\), the standard normal distribution function. It is also known that the analogous convergence relation fails for \(d_k = 1\) (ordinary averages). In this paper we give a fairly complete solution of the problem for which weight sequences the above convergence relation and its refinements hold.

scholarly journals Absorption probabilities for Gaussian polytopes and regular spherical simplices

2020 ◽  
Vol 52 (2) ◽  
pp. 588-616
Author(s):  
Zakhar Kabluchko ◽  
Dmitry Zaporozhets
Keyword(s):  
Distribution Function ◽  
Main Tool ◽  
Standard Normal Distribution ◽  
Normal Vector ◽  
Normal Distribution Function ◽  
Standard Normal Distribution Function ◽  
Expected Number ◽  
Standard Normal ◽  
Crofton Formula ◽  
Absorption Probabilities

AbstractThe Gaussian polytope $\mathcal P_{n,d}$ is the convex hull of n independent standard normally distributed points in $\mathbb{R}^d$ . We derive explicit expressions for the probability that $\mathcal P_{n,d}$ contains a fixed point $x\in\mathbb{R}^d$ as a function of the Euclidean norm of x, and the probability that $\mathcal P_{n,d}$ contains the point $\sigma X$ , where $\sigma\geq 0$ is constant and X is a standard normal vector independent of $\mathcal P_{n,d}$ . As a by-product, we also compute the expected number of k-faces and the expected volume of $\mathcal P_{n,d}$ , thus recovering the results of Affentranger and Schneider (Discr. and Comput. Geometry, 1992) and Efron (Biometrika, 1965), respectively. All formulas are in terms of the volumes of regular spherical simplices, which, in turn, can be expressed through the standard normal distribution function $\Phi(z)$ and its complex version $\Phi(iz)$ . The main tool used in the proofs is the conic version of the Crofton formula.

ON THE GENERALISATION OF SIDEL’NIKOV’S THEOREM TO -ARY LINEAR CODES

2019 ◽  
Vol 101 (1) ◽  
pp. 157-162
Author(s):  
YILUN WEI ◽  
BO WU ◽  
QIJIN WANG
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Linear Codes ◽  
Cumulative Distribution ◽  
Standard Normal Distribution ◽  
Binary Codes ◽  
Normal Distribution Function ◽  
Code Length ◽  
Standard Normal Distribution Function ◽  
Standard Normal

We generalise Sidel’nikov’s theorem from binary codes to $q$-ary codes for $q>2$. Denoting by $A(z)$ the cumulative distribution function attached to the weight distribution of the code and by $\unicode[STIX]{x1D6F7}(z)$ the standard normal distribution function, we show that $|A(z)-\unicode[STIX]{x1D6F7}(z)|$ is bounded above by a term which tends to $0$ when the code length tends to infinity.

scholarly journals Calculation of the throwing angle of a fertilizer centrifugal device as functions of random coordinates of feeding point

2020 ◽  
Vol 175 ◽  
pp. 05016 ◽  
Author(s):  
Vasiliy Chernovolov ◽  
Lyudmila Kravchenko ◽  
Alla Nikitina ◽  
Vladimir Litvinov
Keyword(s):  
Distribution Function ◽  
Normal Distribution ◽  
Angular Speed ◽  
Standard Normal Distribution ◽  
Normal Distribution Function ◽  
Angle Distribution ◽  
Particle Movement ◽  
The Matrix ◽  
Standard Normal ◽  
Centrifugal Device

Liquid fertilizers fed into centrifugal device are spread along angle and radius of feed under action of blades. This article describes how to calculate throwing angular characteristics using Mathcad. The package consists of four programs. Program Mf is intended for calculation of probability density of supply point coordinates under assumption of bivariant normal distribution of system r, γ, which are specified in the form of vectors. The result of the calculation is displayed as matrix Mf. The program Mα calculates the throwing angle for all combinations r, γ.. To calculate the throwing angle, the method of solving differential equations of particle movement along the blade of the device with input data was used: Radius of the disk R, angular speed ω, coefficient of friction of fertilizers on the blade f. The program Ms extracts from the matrix Mf elements Corresponding to a throw angle less than a given number A. The program F (A) sums the elements of the matrix Ms. We obtained the values of the throw angle distribution function by multiplying the resulting sum by the intervals of vectors r and γ. The calculated throwing angle distribution function is approximated by the standard normal distribution function.

scholarly journals High Accurate Simple Approximation of Normal Distribution Integral

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Hector Vazquez-Leal ◽  
Roberto Castaneda-Sheissa ◽  
Uriel Filobello-Nino ◽  
Arturo Sarmiento-Reyes ◽  
Jesus Sanchez Orea
Keyword(s):  
Distribution Function ◽  
Normal Distribution ◽  
Relative Error ◽  
Gaussian Distribution ◽  
Error Function ◽  
Cumulative Distribution ◽  
Standard Normal Distribution ◽  
Normal Distribution Function ◽  
Standard Normal ◽  
Gaussian Integral

The integral of the standard normal distribution function is an integral without solution and represents the probability that an aleatory variable normally distributed has values between zero and . The normal distribution integral is used in several areas of science. Thus, this work provides an approximate solution to the Gaussian distribution integral by using the homotopy perturbation method (HPM). After solving the Gaussian integral by HPM, the result served as base to solve other integrals like error function and the cumulative distribution function. The error function is compared against other reported approximations showing advantages like less relative error or less mathematical complexity. Besides, some integrals related to the normal (Gaussian) distribution integral were solved showing a relative error quite small. Also, the utility for the proposed approximations is verified applying them to a couple of heat flow examples. Last, a brief discussion is presented about the way an electronic circuit could be created to implement the approximate error function.

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