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Local limit theorems for the maxima of discrete random variables

Published online by Cambridge University Press:  24 October 2008

C. W. Anderson
C. W. Anderson
Affiliation:
University of Sheffield
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Extract

Let , where the Xi, i = 1, 2, … are independent identically distributed random variables. Classical extreme value theory, described for example in the books of do Haan(6) and Galambos(3) gives conditions under which there exist constants an > 0 and bn such that

where G(x) is taken to be one of the extreme value distributions G1(x) = exp (− e−x), G2(x) = exp (− x−a) (x > 0, α > 0) and G3(x) = exp (−(− x)α) (x < 0, α > 0).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 88 , Issue 1 , July 1980 , pp. 161 - 165
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

REFERENCES

(1)Anderson, C. W.Extreme value theory for a class of discrete distributions with applications to some stochastic processes. J. Appl. Prob. 7 (1970), 99–113.CrossRefGoogle Scholar
(2)Anderson, C. W. Contributions to the asymptotic theory of extreme values (Ph.D. thesis London University, 1971).Google Scholar
(3)Galambos, J.The asymptotic theory of extreme order statistics (New York. Wiley, 1978).Google Scholar
(4)Gardiner, D. J. and Kimber, A. C. A numerical investigation of the convergence of the maximum of a set of independent Poisson variables to the extreme 2-point distribution (Research Report 193, Manchester-Sheffield School of Probability and Statistics, 1978).Google Scholar
(5)Gnedenko, B. V.Sur la distribution limite du terme maximum d'une serie aléatoire. Ann. Math. 44 (1943), 423–453.CrossRefGoogle Scholar
(6) de Haan, L.On regular variation and its application to the weak convergence of sample extremes (Amsterdam, Mathematisch Centrum, 1970).Google Scholar
(7)Pickands, J.III. Sample sequences of maxima. Ann. Math. Stat. 38 (1967), 1570–1574.Google Scholar