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Limiting behaviour of the distributions of the maxima of partial sums of certain random walks

Published online by Cambridge University Press:  14 July 2016

D. J. Emery
D. J. Emery*
Affiliation:
Polytechnic of Central London
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Abstract

It is shown that, under certain conditions, satisfied by stable distributions, symmetric distributions, distributions with zero mean and finite second moment and other distributions, the distribution function of the maxima of successive partial sums of identically distributed random variables has an asymptotic property. This property implies the regular variation of the tail of the distribution of the hitting times of the associated random walk, and hence that these hitting times belong to the domain of attraction of a stable law.

Keywords

INDEPENDENT IDENTICALLY DISTRIBUTED RANDOM VARIABLESSLOWLY VARYINGREGULARLY VARYINGDOMAIN OF ATTRACTIONSTABLE LAWTAUBERIAN THEOREM
Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 3 , June 1972 , pp. 572 - 579
Copyright
Copyright © Applied Probability Trust 1972 

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References

[1] Gnedenko, B. V. and Kolmogorov, A. N. (1954) Limit Distributions for Sums of Independent Random Variables. Addis on-Wesley Publishing Co., Reading, Mass.Google Scholar
[2] Heyde, C. C. (1969) On the maximum of sums of random variables and the supremum functional for stable processes. J. Appl. Prob. 6, 419–429.Google Scholar
[3] Spitzer, F. (1956) A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82, 323–339.Google Scholar
[4] Hardy, G. H. (1949) Divergent Series. Clarendon Press, Oxford.Google Scholar
[5] Feller, W. (1965) An Introduction to Probability Theory and its Applications. Vol. 2. J. Wiley, New York.Google Scholar
[6] Rosén, B. (1962) On the asymptotic distribution of sums of independent identically distributed random variables. Ark. Mat. 4, 323–332.Google Scholar
[7] Spitzer, F. (1964) Principles of Random Walk. D. Van Nostrand Co. Ltd., Princeton, New Jersey.Google Scholar