Discussiones 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
e-mail: J.Misiewicz@im.uz.zgora.pl

Roger Cooke

Facultteit Informatietechnologie en System, Technische Universiteit Delft
Mekelweg 4, Postbus 5031, 2600 GA Delft, Holland
e-mail: r.m.cooke@its.tudelft.hl

Abstract

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 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.

Keywords: measure, convolution of measures, characteristic function, simple fraction.

2000 Mathematics Subject Classification: 60A10, 60B99.

References

[1] W. Feller, An Introduction to Probability Theory and its Application, volume II. Wiley, New York 1966.
[2] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, Fifth Edition, Academic Press 1997.
[3] N. Jacobson, Basic Algebra I, W.H. Freeman and Company, San Francisco 1974.
[4] S. Lang, Algebra, Addison-Weslay 1970, Reading USA, Second Edition.

Received 10 January 2002