IDEAS home Printed from
   My bibliography  Save this paper

Sums and Extreme Values of Random Variables: Duality Properties


  • Ralph W. Bailey


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.

Suggested Citation

  • Ralph W. Bailey, 2009. "Sums and Extreme Values of Random Variables: Duality Properties," Discussion Papers 09-05, Department of Economics, University of Birmingham.
  • Handle: RePEc:bir:birmec:09-05

    Download full text from publisher

    File URL:
    Download Restriction: no

    References listed on IDEAS

    1. DiNardo, John & Fortin, Nicole M & Lemieux, Thomas, 1996. "Labor Market Institutions and the Distribution of Wages, 1973-1992: A Semiparametric Approach," Econometrica, Econometric Society, vol. 64(5), pages 1001-1044, September.
    2. Checchi, Daniele & Visser, Jelle & van de Werfhorst, Herman G., 2007. "Inequality and Union Membership: The Impact of Relative Earnings Position and Inequality Attitudes," IZA Discussion Papers 2691, Institute for the Study of Labor (IZA).
    3. Angrist, Joshua D. & Krueger, Alan B., 1999. "Empirical strategies in labor economics," Handbook of Labor Economics,in: O. Ashenfelter & D. Card (ed.), Handbook of Labor Economics, edition 1, volume 3, chapter 23, pages 1277-1366 Elsevier.
    4. Bell, Brian D & Pitt, Michael K, 1998. "Trade Union Decline and the Distribution of Wages in the UK: Evidence from Kernel Density Estimation," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 60(4), pages 509-528, November.
    5. David Card & Richard B. Freeman, 1993. "Small Differences That Matter: Labor Markets and Income Maintenance in Canada and the United States," NBER Books, National Bureau of Economic Research, Inc, number card93-1, January.
    6. Richard B. Freeman, 2005. "What Do Unions Do?-- The 2004 M-Brane Stringtwister Edition," Journal of Labor Research, Transaction Publishers, vol. 26(4), pages 641-668, November.
    7. repec:eme:rleczz:s0147-9121(06)26009-6 is not listed on IDEAS
    8. David Card & Thomas Lemieux & W. Craig Riddell, 2004. "Unions and Wage Inequality," Journal of Labor Research, Transaction Publishers, vol. 25(4), pages 519-562, October.
    9. Richard B. Freeman, 1980. "Unionism and the Dispersion of Wages," ILR Review, Cornell University, ILR School, vol. 34(1), pages 3-23, October.
    10. Machin, Stephen, 1997. "The decline of labour market institutions and the rise in wage inequality in Britain," European Economic Review, Elsevier, vol. 41(3-5), pages 647-657, April.
    11. Juan Francisco Canal Domínguez & César Rodríguez Gutiérrez, 2004. "Collective Bargaining and Within-firm Wage Dispersion in Spain," British Journal of Industrial Relations, London School of Economics, vol. 42(3), pages 481-506, September.
    12. Card, David, 1996. "The Effect of Unions on the Structure of Wages: A Longitudinal Analysis," Econometrica, Econometric Society, vol. 64(4), pages 957-979, July.
    13. David Card, 2001. "The Effect of Unions on Wage Inequality in the U.S. Labor Market," ILR Review, Cornell University, ILR School, vol. 54(2), pages 296-315, January.
    14. Amanda Gosling & Thomas Lemieux, 2004. "Labor Market Reforms and Changes in Wage Inequality in the United Kingdom and the United States," NBER Chapters,in: Seeking a Premier Economy: The Economic Effects of British Economic Reforms, 1980-2000, pages 275-312 National Bureau of Economic Research, Inc.
    15. Addison, John T. & Bailey, Ralph & Siebert, W. Stanley, 2003. "The Impact of Deunionisation on Earnings Dispersion Revisited," IZA Discussion Papers 724, Institute for the Study of Labor (IZA).
    Full references (including those not matched with items on IDEAS)

    More about this item


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

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C16 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Econometric and Statistical Methods; Specific Distributions

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. 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). General contact details of provider: .

    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.

    We have no references for this item. You can help adding them by using 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 RePEc Author Service 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.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.