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On the variance of the maximum of partial sums of n-exchangeable random variables with applications

Published online by Cambridge University Press:  14 July 2016

A. A. Anis  and
M. Gharib
A. A. Anis
Affiliation:
Ein Shams University, Cairo
M. Gharib*
Affiliation:
Ein Shams University, Cairo
*
∗∗Postal address: Ein Shams University, Kasr-el-Zaafran, Abbasiya, Cairo, Egypt.
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Abstract

A general formula is obtained for the variance of the maximum of partial sums of n-exchangeable random variables, derived from a result of Spitzer's. The formula is applied in particular to obtain the variance of the maximum of adjusted rescaled partial sums of normal summands. This is of direct relevance to the Hurst effect.

Keywords

VARIANCEPARTIAL SUMSMAXIMUMEXCHANGEABLENORMAL VARIATESSPITZER'S IDENTITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 2 , June 1980 , pp. 432 - 439
Copyright
Copyright © Applied Probability Trust 1980 

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Footnotes

∗

Present address: Arab Planning Institute-Kuwait, P.O. Box 5834, Safad, Kuwait.

References

[1] Anis, A. A. and Lloyd, E. H. (1976) The expected value of the adjusted rescaled Hurst range of independent normal summands. Biometrika 63, 111–116.Google Scholar
[2] Anis, A. A. and Lloyd, E. H. (1977) On the distribution of the Hurst range of independent normal summands. Research report, International Institute for Applied Systems Analysis, Vienna, July 1977.Google Scholar
[3] 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
[4] Kac, M. (1954) Toeplitz matrices, translation kernels and a related problem in probability theory. Duke Math. J. 21, 501–509.CrossRefGoogle Scholar
[5] Solari, M. E. and Anis, A. A. (1957) The mean and variance of the maximum of the adjusted partial sums of a finite number of independent normal variates. Ann. Math. Statist. 28, 706–716.Google Scholar
[6] Spitzer, F. (1956) A combinatorial lemma and its applications to probability theory. Trans. Amer. Math. Soc. 82, 323–339.Google Scholar