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About the density of spectral measure of the two-dimensional SaS random vector

100%

Borowiecka-Olszewska M., Misiewicz J.

Discussiones Mathematicae Probability and Statistics

2003

tom 23

nr 1

77-81

In this paper, we consider a symmetric α-stable p-sub-stable two-dimensional random vector. Our purpose is to show when the function $exp{-(|a|p + |b|p)^{α/p}}$ is a characteristic function of such a vector for some p and α. The solution of this problem we can find in [3], in the language of isometric embeddings of Banach spaces. Our proof is based on simple properties of stable distributions and some characterization given in [4].

Simple fractions and linear decomposition of some convolutions of measures

100%

Misiewicz J., Cooke R.

Discussiones Mathematicae Probability and Statistics

2001

tom 21

nr 2

149-157

Every characteristic function φ can be written in the following way: φ(ξ) = 1/(h(ξ) + 1), where h(ξ) = ⎧ 1/φ(ξ) - 1 if φ(ξ) ≠ 0 ⎨ ⎩ ∞ if φ(ξ) = 0 This simple remark implies that every characteristic function can be treated as a simple fraction of the function h(ξ). In the paper, we consider a class C(φ) of all characteristic functions of the form $φ_{a}(ξ) = [a/(h(ξ) + a)]$, where φ(ξ) is a fixed characteristic function. Using the well known theorem on simple fraction decomposition of rational functions we obtain that convolutions of measures $μ_{a}$ with $μ̂_{a}(ξ) = φ_{a}(ξ)$ are linear combinations of powers of such measures. This can simplify calculations. It is interesting that this simplification uses signed measures since coefficients of linear combinations can be negative numbers. All the results of this paper except Proposition 1 remain true if we replace probability measures with complex valued measures with finite variation, and replace the characteristic function with Fourier transform.

Ograniczanie wyników

2 Discussiones Mathematicae Probability and Statistics

2 Misiewicz J.

1 Borowiecka-Olszewska M.

1 Cooke R.

1 2003

1 2001

Wyniki wyszukiwania

About the density of spectral measure of the two-dimensional SaS random vector

Simple fractions and linear decomposition of some convolutions of measures