Sums and Extreme Values of Random Variables: Duality Properties
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.
|Date of creation:||Jun 2009|
|Date of revision:|
|Contact details of provider:|| Postal: |
Web page: http://www.economics.bham.ac.uk
More information through EDIRC
When requesting a correction, please mention this item's handle: RePEc:bir:birmec:09-05. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Colin Rowat)
If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.
If references are entirely missing, you can add them using this form.
If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.
If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.
Please note that corrections may take a couple of weeks to filter through the various RePEc services.