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Order statistics of partial sums of mutually independent random variables

Published online by Cambridge University Press:  14 July 2016

Felix Pollaczek
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Abstract

Herein is exposed a simplified analytic proof of formulas for the characteristic functions of ordered partial sums of mutually independent identically distributed random variables. This problem which we had raised and solved in 1952 by another method, has since been treated by several authors (see Wendel [6]), and recently by de Smit [4], who made use of a kind of Wiener-Hopf decomposition†. On the contrary our present as well as our previous proof essentially uses the explicit solution of a certain singular integral equation in a complex domain.

Keywords

ORDER STATISTICSRANDOM VARIABLESSTOCHASTIC INDEPENDENCESINGULAR LINEAR INTEGRAL EQUATIONSJOINT PROBABILITIESJOINT CHARACTERISTIC FUNCTIONS
Type
Short Communications
Information
Journal of Applied Probability , Volume 12 , Issue 2 , June 1975 , pp. 390 - 395
Copyright
Copyright © Applied Probability Trust 1975 

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Footnotes

† De Smit's remark in the summary of [4], that my results generalise F. Spitzer's identity, is due to a chronological error. Using a different method, Spitzer treated in 1956 a particular case of my problem of 1952.

References

[1] 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
[2] Pollaczek, F., (1957) Problèmes stochastiques posés par le phénomène de formation d'une queue d'attente à un guichet et par des phénomènes apparentés. Mémor. Sci. Math. 136, 1–123.Google Scholar
[3] Pollaczek, F., (1961) Théorie analytique des problèmes stochastiques relatifs á un groupe de lignes téléphoniques avec dispositif d'attente. Mémor. Sci. Math. 150, 1–115.Google Scholar
[4] De Smit, J. H. A. (1973) A simple analytic proof of the Pollaczek-Wendel identity for ordered partial sums. Ann. of Prob. 1, 348–351.Google Scholar
[5] Spitzer, F. (1956) A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82, 323–339.Google Scholar
[6] Wendel, J. G. (1960) Order statistics of partial sums. Ann. Math. Statist. 31, 1034–1044.Google Scholar