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Sums and Extreme Values of Random Variables: Duality Properties

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  • Ralph W. Bailey

Abstract

The inversion theorem for radially-distributed complex random variables provides a completely symmetric relationship between their characteristic functions and their distribution functions, suitably defi- ?ned. If the characteristic function happens also to be a distribution function, then a dual pair of random variables is de?fined. The distrib- ution function of each is the characteristic function of the other. If we call any distribution possessing a dual partner 'invertible', then both the radial normal and radial t distributions are invertible. Moreover the product of an invertible variable (for instance, a radial normal variable) with any other independent variable is invertible. Though the most prominent examples of invertible variables possess a normal divisor, we exhibit a pair of variables neither of which has a normal di- visor. A test for normal-divisibility, based on complete monotonicity, is provided. The sum of independent invertible variables is invertible; the inverse is the smallest in magnitude of the inverse variables. The- orems about sums of invertible random variables (for instance, central limit theorems) have a dual interpretation as theorems about extrema, and vice versa.

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Bibliographic Info

Paper provided by Department of Economics, University of Birmingham in its series Discussion Papers with number 09-05.

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Length: 11 pages
Date of creation: Jun 2009
Date of revision:
Handle: RePEc:bir:birmec:09-05

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Related research

Keywords: Bernstein's theorem; Bessel transform; duality; extreme value theorem; radial distribution; t-distribution;

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