A functional equation and its application to the characterization of the Weibull and stable distributions

1976 ◽  
Vol 13 (2) ◽  
pp. 385-391 ◽  
Author(s):  
Y. H. Wang
Keyword(s):  
Functional Equation ◽  
Weibull Distribution ◽  
Exponential Distribution ◽  
Stable Distribution ◽  
Stable Distributions ◽  
Cauchy Functional Equation ◽  
Linear Statistics

The Cauchy functional equation Φ(x + y) = Φ(x) + Φ(y) is generalized to the form , assuming Φ is left- or right- continuous. This result is used to obtain (1) a characterization of the Weibull distribution, in the spirit of the memoryless property of the exponential distribution, by , for all x, y ≧ 0;(2) a characterization of the symmetric α-stable distribution by the equidistribution of linear statistics.

A functional equation and its application to the characterization of the Weibull and stable distributions

1976 ◽  
Vol 13 (02) ◽  
pp. 385-391
Author(s):  
Y. H. Wang
Keyword(s):  
Functional Equation ◽  
Weibull Distribution ◽  
Exponential Distribution ◽  
Stable Distribution ◽  
Stable Distributions ◽  
Cauchy Functional Equation ◽  
Linear Statistics

The Cauchy functional equation Φ(x+y) = Φ(x) + Φ(y) is generalized to the form, assuming Φ is left- or right- continuous. This result is used to obtain (1) a characterization of the Weibull distribution, in the spirit of the memoryless property of the exponential distribution, by, for allx,y≧ 0;(2) a characterization of the symmetricα-stable distribution by the equidistribution of linear statistics.

scholarly journals On a conditional Cauchy functional equation of several variables and a characterization of multivariate stable distributions

1993 ◽  
Vol 16 (1) ◽  
pp. 165-168 ◽  
Author(s):  
Arjun K. Gupta ◽  
Truc T. Nguyen ◽  
Wei-Bin Zeng
Keyword(s):  
Functional Equation ◽  
General Solution ◽  
Stable Distributions ◽  
Cauchy Functional Equation ◽  
Several Variables

The general solution of a conditional Cauchy functional equation of several variables is obtained and its applications to the characterizations of multivariate stable distributions are studied.

scholarly journals Some results concerning the Cauchy functional equation in certain Banach algebras

1985 ◽  
Vol 31 (1) ◽  
pp. 137-144 ◽  
Author(s):  
J. Vukman
Keyword(s):  
Functional Equation ◽  
Hilbert Space ◽  
Identity Element ◽  
Banach Algebras ◽  
Cauchy Functional Equation

In this paper some results concerning the Cauchy functional equation, that is the functional equation f(x+y) = f(x) + f(y) in complex hermitian Banach *-algebras with an identity element are presented. As an application a generalization of Kurepa's extension of the Jordan-Neumann characterization of pre-Hilbert space is obtained.

scholarly journals A direct approach to the stable distributions

2016 ◽  
Vol 48 (A) ◽  
pp. 261-282 ◽  
Author(s):  
E. J. G. Pitman ◽  
Jim Pitman
Keyword(s):  
Functional Equation ◽  
Characteristic Function ◽  
Integral Representation ◽  
Explicit Form ◽  
Regular Variation ◽  
Stable Distribution ◽  
Stable Distributions ◽  
Direct Approach ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible

AbstractThe explicit form for the characteristic function of a stable distribution on the line is derived analytically by solving the associated functional equation and applying the theory of regular variation, without appeal to the general Lévy‒Khintchine integral representation of infinitely divisible distributions.

A Characterization of Exponential Functions with Non-Linear Exponents

1975 ◽  
Vol 18 (2) ◽  
pp. 277-281
Author(s):  
Halina Światak
Keyword(s):  
Functional Equation ◽  
Cauchy Functional Equation ◽  
Exponential Functions ◽  
Non Linear ◽  
Image Position

It is well known (see e.g. [1]) that the Cauchy functional equationcharacterizes the function f:x→exx.It was mentioned in [2D that the functions f:x→A exp(αx2m), g:x→exx/A can be characterized by the equation(1)but the proof was done only for m=l which was considerably simple.

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