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A note on Spitzer's lemma and its application to the maximum of partial sums of dependent random variables

Published online by Cambridge University Press:  14 July 2016

M. E. Solari  and
J. E. A. Dunnage
M. E. Solari*
Affiliation:
Chelsea College, London
J. E. A. Dunnage
Affiliation:
Chelsea College, London
*
† Address: Oakside, Duffield Lane, Stoke Poges, Slough, Bucks. SL2 4AH.
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Abstract

We give an expression for the expectation of max (0, S1, …, Sn) where Sk is the kth partial sum of a finite sequence of exchangeable random variables X1, …, Xn. When the Xk are also independent, the formula we give has already been obtained by Spitzer; and when the sequence is a finite segment of an infinite sequence of exchangeable random variables, it is a consequence of a theorem of Hewitt.

Keywords

DEPENDENT RANDOM VARIABLESEXCHANGEABLE RANDOM VARIABLESMAXIMUM PARTIAL SUMSPITZER'S LEMMA
Type
Short Communications
Information
Journal of Applied Probability , Volume 13 , Issue 2 , June 1976 , pp. 361 - 364
Copyright
Copyright © Applied Probability Trust 1976 

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Footnotes

*

In this note equations and theorems are numbered as in Spitzer (1956).

References

Boes, D. C. and Salas-La Cruz, J. D. (1973) On the expected range and the expected adjusted range of partial sums of exchangeable random variables. J. Appl. Prob. 10, 671–677.Google Scholar
Spitzer, F. (1956) A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82, 323–339.Google Scholar