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.

Bounds for the Characteristic Functions of the System of Monomials in Random Variables and of Its Trigonometric Analogue

1997 ◽  
Vol 4 (6) ◽  
pp. 579-584
Author(s):  
T. Shervashidze
Keyword(s):  
Random Vector ◽  
Continuous Distribution ◽  
Random Variables ◽  
Upper Bounds ◽  
Characteristic Functions ◽  
Absolutely Continuous ◽  
Multidimensional Analogue ◽  
Trigonometric Integral

Abstract Using a multidimensional analogue of Vinogradov's inequality for a trigonometric integral, the upper bounds are constructed for the moduli of the characteristic functions both of the system of monomials in components of a random vector with an absolutely continuous distribution in and of the system (cos j 1πξ 1 . . . cos j s πξ s , 0 ≤ j 1, . . . , j s ≤ k, j 1 + . . . + j s ≥ 1), where (ξ 1, . . . , ξ s ) is uniformly distributed in [0; 1] s .

Recurrence for Markov processes on N lines

1971 ◽  
Vol 8 (04) ◽  
pp. 724-730
Author(s):  
Mark Pinsky
Keyword(s):  
Markov Process ◽  
Characteristic Function ◽  
Markov Processes ◽  
Real Line ◽  
Transition Function ◽  
Characteristic Functions ◽  
The Real ◽  
Matrix Logarithm ◽  
The Matrix ◽  
Continuous Markov Process

Let Λ = R 1 × {1, 2, ···, N} denote N copies of the real line and ξ(t) = (X(t), α(t))be a right-continuous Markov process taking values in A having transition function of the form P(t, (x, α), A × {β}) = Fαβ (t, A – x). Fukushima and Hitsuda [2] have found the most general such transition function; the (matrix) logarithm of its characteristic function is decomposed into a Lévy-Khintchine integral on the diagonal and multiples of characteristic functions off the diagonal.

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.

Recurrence for Markov processes on N lines

1971 ◽  
Vol 8 (4) ◽  
pp. 724-730
Author(s):  
Mark Pinsky
Keyword(s):  
Markov Process ◽  
Characteristic Function ◽  
Markov Processes ◽  
Real Line ◽  
Transition Function ◽  
Characteristic Functions ◽  
The Real ◽  
Matrix Logarithm ◽  
The Matrix ◽  
Continuous Markov Process

Let Λ = R1 × {1, 2, ···, N} denote N copies of the real line and ξ(t) = (X(t), α(t))be a right-continuous Markov process taking values in A having transition function of the form P(t, (x, α), A × {β}) = Fαβ(t, A – x). Fukushima and Hitsuda [2] have found the most general such transition function; the (matrix) logarithm of its characteristic function is decomposed into a Lévy-Khintchine integral on the diagonal and multiples of characteristic functions off the diagonal.

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.

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.

scholarly journals The Algebraic Degree of Phase-Type Distributions

2010 ◽  
Vol 47 (03) ◽  
pp. 611-629
Author(s):  
Mark Fackrell ◽  
Qi-Ming He ◽  
Peter Taylor ◽  
Hanqin Zhang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Polynomial Algorithm ◽  
Algebraic Degree ◽  
Nonnegative Matrices ◽  
Stieltjes Transform ◽  
Special Cases ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Matrix

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

Certain Fourier Transforms of Distributions: II

1954 ◽  
Vol 6 ◽  
pp. 186-189 ◽  
Author(s):  
Eugene Lukacs ◽  
Otto Szász
Keyword(s):  
Characteristic Function ◽  
Fourier Transforms ◽  
Necessary Conditions ◽  
Laplace Transforms ◽  
Characteristic Functions ◽  
Necessary Condition ◽  
Multiple Roots ◽  
Not Given ◽  
Class Of Functions

In an earlier paper (1), published in this journal, a necessary condition was given which the reciprocal of a polynomial without multiple roots must satisfy in order to be a characteristic function. This condition is, however, valid for a wider class of functions since it can be shown (2, theorem 2 and corollary to theorem 3) that it holds for all analytic characteristic functions. The proof given in (1) is elementary and has some methodological interest since it avoids the use of theorems on singularities of Laplace transforms. Moreover the method used in (1) yields some additional necessary conditions which were not given in (1) and which do not seem to follow easily from the properties of analytic characteristic functions.

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