Note on a relation between the distribution functions and characteristic functions

1954 ◽  
Vol 6 (3) ◽  
pp. 191-195
Author(s):  
Minoru Motoo
Keyword(s):  
Distribution Functions ◽  
Characteristic Functions

Matrix variate pareto distributions

2021 ◽  
Vol 71 (2) ◽  
pp. 475-490
Author(s):  
Shokofeh Zinodiny ◽  
Saralees Nadarajah
Keyword(s):  
Distribution Functions ◽  
Cumulative Distribution ◽  
Characteristic Functions ◽  
Type I ◽  
Type Ii ◽  
Cumulative Distribution Functions ◽  
Pareto Distributions

Abstract Matrix variate generalizations of Pareto distributions are proposed. Several properties of these distributions including cumulative distribution functions, characteristic functions and relationship to matrix variate beta type I and matrix variate type II distributions are studied.

Certain Fourier Transforms of Distributions

1951 ◽  
Vol 3 ◽  
pp. 140-144 ◽  
Author(s):  
Eugene Lukacs ◽  
Otto Szasz
Keyword(s):  
Probability Distribution ◽  
Characteristic Function ◽  
Fourier Transforms ◽  
Probability Distributions ◽  
Distribution Functions ◽  
Characteristic Functions

Fourier transforms of distribution functions are frequently studied in the theory of probability. In this connection they are called characteristic functions of probability distributions. It is often of interest to decide whether a given function φ(t) can be the characteristic function of a probability distribution, that is, whether it admits the representation

Aspects of the modelling of the radial distribution function for small nanoparticles

2007 ◽  
Vol 40 (6) ◽  
pp. 975-985 ◽  
Author(s):  
Vladimir I. Korsunskiy ◽  
Reinhard B. Neder ◽  
Andreas Hofmann ◽  
Sofia Dembski ◽  
Christina Graf ◽  
...  
Keyword(s):  
Radial Distribution ◽  
High Energy ◽  
Relative Magnitude ◽  
Distribution Functions ◽  
Characteristic Functions ◽  
Znse Nanoparticles ◽  
Interatomic Distances ◽  
Angle Scattering ◽  
Wide Range ◽  
Spherical Core

An approach to modelling radial distribution functions (RDFs) of nanoparticle samples over a wide range of interatomic distances is presented. Two different types of contribution to the model RDF are calculated. The first explicitly reflects the structure of the nanoparticle parts with more or less crystalline atomic structure. It can be calculated precisely and contains comparatively sharp peaks, which are produced by the set of discrete interatomic distances. The second includes RDF contributions from distances between weakly correlated atoms positioned within different nanoparticles or within different parts of a nanoparticle model. The calculation is performed using the approximation of a uniform distribution of atoms and utilizes the ideas of the characteristic functions of the particle shape known in small-angle scattering theory. This second RDF contribution is represented by slowly varying functions of interatomic distancer. The relative magnitude of this essential part of the model RDF increases with increasingrcompared with the part that represents the ordered structure. The method is applied to test several spherical and core/shell models of semiconductor nanoparticles stabilized with organic ligands. The experimental RDFs of ZnSe and CdSe/ZnS nanoparticle samples were obtained by high-energy X-ray diffraction at beamline BW5, HASYLAB, DESY. The ZnSe nanoparticles have a spherical core with approximately 26 Å diameter and zincblende structure. The RDF of the CdSe/ZnS nanoparticle sample shows resolved peaks of the first- and the second-neighbour distances characteristic for CdSe (2.62 and 4.27 Å) and for ZnS (2.33 and 3.86 Å) and for the first time clearly confirms the presence of CdSe and ZnS nanophases in such objects. The diameters of the CdSe and ZnS spherical cores are estimated as 27 and 15 Å. CdSe and ZnS are present in the sample for the most part as independent nanoparticles. A smaller amount of ZnS forms an irregularly shaped shell around the CdSe cores, which consists of small independently oriented ZnS particles.

Obtaining Distribution Functions by Numerical Inversion of Characteristic Functions with Applications

1995 ◽  
Vol 49 (4) ◽  
pp. 346 ◽  
Author(s):  
Lance A. Waller ◽  
Bruce W. Turnbull ◽  
J. Michael Hardin
Keyword(s):  
Distribution Functions ◽  
Characteristic Functions ◽  
Numerical Inversion

scholarly journals On convergence and compactness in variation with shift of discrete probability laws

2021 ◽  
Vol 8 (3) ◽  
pp. 385-393
Author(s):  
Ivan A. Alexeev ◽  
◽  
Alexey A. Khartov ◽  
Keyword(s):  
Real Line ◽  
Discrete Distribution ◽  
Distribution Functions ◽  
Divisible Distribution ◽  
Characteristic Functions ◽  
Infinitely Divisible Distribution ◽  
Infinitely Divisible ◽  
The Real ◽  
Convergence In Variation ◽  
Compactness Theorems

We consider a class of discrete distribution functions, whose characteristic functions are separated from zero, i. e. their absolute values are greater than positive constant on the real line. The class is rather wide, because it contains discrete infinitely divisible distribution functions, functions of lattice distributions, whose characteristic functions have no zeroes on the real line, and also distribution functions with a jump greater than 1/2. Recently the authors showed that characteristic functions of elements of this class admit the Lévy-Khinchine type representations with non-monotonic spectral function. Thus our class is included in the set of so called quasi-infinitely divisible distribution functions. Using these representation the authors also obtained limit and compactness theorems with convergence in variation for the sequences from this class. This note is devoted to similar results concerning convergence and compactness but with weakened convergence in variation. Replacing of type of convergence notably expands applicability of the results.

scholarly journals FORMULA OF INVERSION FOR RATIONAL CHARACTERISTIC FUNCTIONS OF PROBABILITY DISTRIBUTIONS

2018 ◽  
pp. 7-22
Author(s):  
Georgiy Aleksandrovich Popov
Keyword(s):  
Distribution Function ◽  
Probability Distributions ◽  
Rational Functions ◽  
Distribution Functions ◽  
Characteristic Functions ◽  
Arbitrary Distribution ◽  
Abscissa Axis ◽  
Entire Axis ◽  
Inversion Formulas ◽  
Gamma Distributions

The paper deals with the problem of clarifying the well-known inversion formulas for distribution functions, usually describing the increment of these functions. The validity of the corresponding inversion formulas for the distribution function π and their densities has been proved for the particular case of distributions with rational characteristic functions. The obtained formulas for distribution functions, which include additionally constant terms equal to 0.5, were not previously known. Functions of positively distributed random variables and quantities distributed over the entire axis have been considered separately. To test the hypothesis of fairness of the obtained treatment formula, including a previously unknown term equal to 0.5, in the general case there have been given examples of calculating distribution functions, whose characteristic functions are not considered as rational functions: for constant and uniform laws. The verification confirmed the objectiveness of the formulated hypothesis about the obtained validity of the inversion form for arbitrary distribution functions. It has also been shown that any distribution function and any density can be represented as a limit of a mixture of gamma distributions (distribution functions and densities), having shifts along the abscissa axis and, possibly, with altered signs of the arguments. The obtained result proves that the set of gamma distributions with shifted arguments is uniformly dense in the set of all distributions.

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