Mathematicae Probability and Statistics 21(2) (2001) 149-157
SIMPLE FRACTIONS AND LINEAR DECOMPOSITION OF SOME CONVOLUTIONS OF MEASURES
| Jolanta K. Misiewicz
Institute of Mathematics, University of Zielona Góra
Podgórna 50, 65-246 Zielona Góra, Poland
Facultteit Informatietechnologie en System, Technische Universiteit Delft
Mekelweg 4, Postbus 5031, 2600 GA Delft, Holland
Every characteristic function φ can be written in the following way:
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 φa(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 μa
with [^(μa)](ξ) = φa
(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.
|, where h(ξ) =
Keywords: measure, convolution of measures, characteristic function, simple fraction.
2000 Mathematics Subject Classification: 60A10, 60B99.
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Received 10 January 2002