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Tail equivalence and its applications

Published online by Cambridge University Press:  14 July 2016

Sidney I. Resnick
Sidney I. Resnick*
Affiliation:
Purdue University
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Abstract

If for two c.d.f.'s F(·) and G(·), 1 – F(x)/1 – G(x) → A, 0 <A <∞, as x → ∞, then for normalizing constants an > 0, bn, n > 1, Fn(anx + bn) → φ(x), φ(x) non-degenerate, iff Gn(anx + bn)→ φ A−1(x). Conversely, if Fn(anx+bn)→ φ(x), Gn(anx + bn) → φ'(x), φ(x) and φ'(x) non-degenerate, then there exist constants C >0 and D such that φ'(x) =φ(Cx + D) and limx→∞ 1 — F(x)/1 — G(x) exists and is expressed in terms of C and D, depending on which type of extreme value distribution φ(x) is. These results are used to study domain of attraction questions for products of distribution functions and to reduce the limit law problem for maxima of a sequence of random variables defined on a Markov chain (M.C.) to the independent, identically distributed (i.i.d.) case.

Type
Research Papers
Information
Journal of Applied Probability , Volume 8 , Issue 1 , March 1971 , pp. 136 - 156
Copyright
Copyright © Applied Probability Trust 1971 

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References

[1] Barndorff-Nielsen, O. (1964) On the limit distribution of the maximum of a random number of independent random variables. Acta Math. Acad. Sci. Hung. 15, 399–403.CrossRefGoogle Scholar
[2] Chung, K. L. (1960) Markov Chains with Stationary Transition Probabilities. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[3] Feller, W. (1966) An Introduction to Probability Theory and Its Applications. Vol. 2, Wiley, New York.Google Scholar
[4] Gnedenko, B.V. (1943) Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math. 44, 423–453.CrossRefGoogle Scholar
[5] Resnick, S. I. and Neuts, M. F. (1970) Limit laws for maxima of a sequence of random variables defined on a Markov chain. Adv. Appl. Prob. 2, 323–343.Google Scholar
[6] Resnick, S. I. (1970) Maxima of a Sequence of Random Variables Defined on a Markov Chain. Ph.D. Thesis. Purdue University, Lafayette, Indiana.Google Scholar
[7] Varga, R.S. (1962) Matrix Iterative Analysis. Prentice-Hall, New Jersey.Google Scholar