Characteristic Functions

2021 ◽  
pp. 213-234
Author(s):  
James Davidson
Keyword(s):  
Number Theory ◽  
Characteristic Function ◽  
Complex Number ◽  
Random Variables ◽  
Infinite Divisibility ◽  
Characteristic Functions ◽  
Conditional Distributions ◽  
Inversion Theorem ◽  
Sums Of Random Variables

This chapter begins with a look at convolutions and the distribution of sums of random variables. It briefly surveys complex number theory before defining the characteristic function and studying its properties, with a range of examples. The concept of infinite divisibility is introduced. The important inversion theorem is treated and finally consideration is given to characteristic functions in conditional distributions.

scholarly journals On Bounds for the Characteristic Functions of Some Degenerate Multidimensional Distributions

2003 ◽  
Vol 10 (2) ◽  
pp. 353-362
Author(s):  
T. Shervashidze
Keyword(s):  
Lower Bound ◽  
Characteristic Function ◽  
Chebyshev Polynomials ◽  
Random Variables ◽  
Characteristic Functions ◽  
Multidimensional Analogue ◽  
The Matrix ◽  
Trigonometric Integral

Abstract We discuss an application of an inequality for the modulus of the characteristic function of a system of monomials in random variables to the convergence of the density of the corresponding system of the sample mixed moments. Also, we consider the behavior of constants in the inequality for the characteristic function of a trigonometric analogue of the above-mentioned system when the random variables are independent and uniformly distributed. Both inequalities were derived earlier by the author from a multidimensional analogue of Vinogradov's inequality for a trigonometric integral. As a byproduct the lower bound for the spectrum of is obtained, where 𝐴𝑘 is the matrix of coefficients of the first 𝑘 + 1 Chebyshev polynomials of first kind.

Singular infinitely divisible distributions whose characteristic functions vanish at infinity

1977 ◽  
Vol 82 (2) ◽  
pp. 277-287 ◽  
Author(s):  
Gavin Brown
Keyword(s):  
Dynamical Systems ◽  
Characteristic Function ◽  
Random Variables ◽  
Characteristic Functions ◽  
Strong Mixing ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Full Support ◽  
Singular Spectrum

In the course of discussing dynamical systems which enjoy strong mixing but have singular spectrum, E. Hewitt and the author, (2), recently constructed families of symmetric random variables which satisfy inter alia the following properties:(i) Zt is purely singular and has full support,(ii) χ(t)(x) → 0 as ± x → ∞, where χ(t) is the characteristic function of Zt,(iii)′ Zt+s, Zt + Zs have the same null events,(iv) whenever s ≠ t, Zt and Zs + a are mutually singular for every (possibly zero) constant a.

Probability Functions which are Proportional to Characteristic Functions and the Infinite Divisibility of the von Mises Distribution

1975 ◽  
Vol 12 (S1) ◽  
pp. 19-28 ◽  
Author(s):  
Toby Lewis
Keyword(s):  
Continuous Distribution ◽  
Random Variables ◽  
Discrete Distribution ◽  
Infinite Divisibility ◽  
Reciprocal Relationship ◽  
Characteristic Functions ◽  
Von Mises Distribution ◽  
Infinitely Divisible ◽  
Von Mises ◽  
Divisibility Properties

Reciprocal pairs of continuous random variables on the line are considered, such that the density function of each is, to within a norming factor, the characteristic function of the other. The analogous reciprocal relationship between a discrete distribution on the line and a continuous distribution on the circle is also considered. A conjecture is made regarding infinite divisibility properties of such pairs of random variables. It is shown that the von Mises distribution is infinitely divisible for sufficiently small values of the concentration parameter.

SOME EXTENSIONS OF A LEMMA OF KOTLARSKI

2012 ◽  
Vol 28 (4) ◽  
pp. 925-932 ◽  
Author(s):  
Kirill Evdokimov ◽  
Halbert White
Keyword(s):  
Characteristic Function ◽  
Characteristic Functions ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Weaker Conditions

This note demonstrates that the conditions of Kotlarski’s (1967, Pacific Journal of Mathematics 20(1), 69–76) lemma can be substantially relaxed. In particular, the condition that the characteristic functions of M, U1, and U2 are nonvanishing can be replaced with much weaker conditions: The characteristic function of U1 can be allowed to have real zeros, as long as the derivative of its characteristic function at those points is not also zero; that of U2 can have an isolated number of zeros; and that of M need satisfy no restrictions on its zeros. We also show that Kotlarski’s lemma holds when the tails of U1 are no thicker than exponential, regardless of the zeros of the characteristic functions of U1, U2, or M.

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