Hostname: page-component-84b7d79bbc-l82ql Total loading time: 0 Render date: 2024-07-30T05:02:59.640Z Has data issue: false hasContentIssue false
  • English
  • Français

On the ordered partial sums of real random variables

Published online by Cambridge University Press:  14 July 2016

Lajos Takács
Lajos Takács*
Affiliation:
Case Western Reserve University, Cleveland, Ohio
Get access Rights & Permissions [Opens in a new window]

Abstract

In 1952 Pollaczek discovered a remarkable formula for the Laplace-Stieltjes transforms of the distributions of the ordered partial sums for a sequence of independent and identically distributed real random variables. In this paper Pollaczek's result is proved in a simple way and is extended for a semi-Markov sequence of real random variables.

Keywords

INDEPENDENT RANDOM VARIABLESSEMI-MARKOV SEQUENCESORDERED PARTIAL SUMSLAPLACE TRANSFORMS
Type
Research Papers
Information
Journal of Applied Probability , Volume 14 , Issue 1 , March 1977 , pp. 75 - 88
Copyright
Copyright © Applied Probability Trust 1977 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Andersen, E. S. (1967) An algebraic treatment of fluctuations of sums of random variables. Proc. 5th Berkeley Symp. Math. Statist. Prob. 2 (1), 423–429.Google Scholar
[2] Baxter, G. (1961) An analytic approach to finite fluctuation problems in probability. J. Anal. Math. 9, 31–70.CrossRefGoogle Scholar
[3] Darling, D. A. (1962) A unified treatment of finite fluctuation problems. Colloquium on Combinatorial Methods in Probability Theory, Matematisk Institut, Aarhus Universitet, Denmark, 3–6.Google Scholar
[4] De Smit, J. H. A. (1973) A simple analytic proof of the Pollaczek-Wendel identity for ordered partial sums. Ann. Prob. 1, 348–351.Google Scholar
[5] Dirichlet, G. L. (1839) Sur une nouvelle méthode pour la détérmination des intégrales multiples. C. R. Acad. Sci. Paris 8, 156–160. [G. Lejeune Dirichlet's Werke, Bd. I. Reimer, Berlin (1889), 375–380. Reprinted (1969) by Chelsea, New York.] Google Scholar
[6] Pollaczek, F. (1949) Application d'opérateurs integro-combinatoires dans la théorie des intégrales multiples de Dirichlet. Ann. Inst. H. Poincaré 11, 113–133.Google Scholar
[7] Pollaczek, F. (1952) Fonctions caractéristiques de certaines répartitions définies au moyen de la notion d'ordre. Application à la théorie des attentes. C. R. Acad. Sci. Paris, 234, 2334–2336.Google Scholar
[8] Pollaczek, F. (1975) Order statistics of partial sums of mutually independent random variables. J. Appl. Prob. 12, 390–395.CrossRefGoogle Scholar
[9] Port, S. C. (1963) An elementary probability approach to fluctuation theory. J. Math. Anal. Appl. 6, 109–151.Google Scholar
[10] Takács, L. (1972) On a linear transformation in the theory of probability. Acta Sci. Math. (Szeged) 33, 15–24.Google Scholar
[11] Wendel, J. G. (1960) Order statistics of partial sums. Ann. Math. Statist. 31, 1034–1044.CrossRefGoogle Scholar