Sums and Extreme Values of Random Variables: Duality Properties

Ralph Bailey ()

Discussion Papers from Department of Economics, University of Birmingham

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.

Keywords: Bernstein's theorem; Bessel transform; duality; extreme value theorem; radial distribution; t-distribution (search for similar items in EconPapers)
JEL-codes: C02 C16 (search for similar items in EconPapers)
Pages: 11 pages
Date: 2009-06
New Economics Papers: this item is included in nep-ecm
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