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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
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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 

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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