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Stochastic compactness and point processes

Part of: Limit theorems Nonparametric inference

Published online by Cambridge University Press:  09 April 2009

L. de Hann  and
S. I. Resnick
L. de Hann
Affiliation:
Econometric Institute Erasmus UniversityRotterdam, Holland
S. I. Resnick
Affiliation:
Department of Statistics Colorado State UniversityFort Collins, Colorado 80523, U.S.A.
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Abstract

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We show that stochastic compactness of partial sums with no normal limit distribution corresponds to stochastic compactness of the point processes generated by the observations so that there exist joint limit distributions for the sample sums and the sample maxima.

MSC classification

Secondary: 60F05: Central limit and other weak theorems 62G30: Order statistics; empirical distribution functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 37 , Issue 3 , December 1984 , pp. 307 - 316
Copyright
Copyright © Australian Mathematical Society 1984

References

Anderson, C. W. (1970), ‘Extreme value theory for a class of discrete distributions with applications to some stochastic process’, J. Appl. Probab. 7, 99–113.CrossRefGoogle Scholar
Billingsley, P. (1968), Convergence of probability measures (Wiley, New York).Google Scholar
Durrett, R. and Resnick, S. (1978), ‘Functional limit theorems for dependent variables’, Ann. Probab. 6, 829–846.CrossRefGoogle Scholar
Feller, W. (1966), ‘On regular variation and local limit theorems’, Proc. Fifth Berkeley Symp. Math. Statist. Probability 2, 373–388.Google Scholar
Goldie, C. (1977), ‘Convergence theorems for empirical Lorenz curves and their inverses’, Adv. in Appl. Probab. 9, 765–791.CrossRefGoogle Scholar
de Haan, L. and Ridder, G. (1979), ‘Stochastic compactness of sample extremes’, Ann. Probab. 7, 290–303.CrossRefGoogle Scholar
Hille, E. and Phillips, R. S. (1948), Functional analysis and semigroups (Amer. Math. Soc. Colloq. Publ., vol. 31).Google Scholar
Ito, K. (1969), Stochastic processes (Lecture Notes, Series no. 16, Mathematisk Institut, Aarhus University).Google Scholar
Lindvall, T. (1973), ‘Weak convergence of probability measures and random functions in the function space D[0, ∞)’, J. Appl. Probab. 10, 109–121.CrossRefGoogle Scholar
Maller, R. A. (1980), ‘A note on domains of partial attraction’, Ann. Probab. 8, 576–583.CrossRefGoogle Scholar
Maller, R. A. (1981), ‘Some properties of stochastic compactness’, J. Austral. Math. Soc. Ser A 30, 263–277.Google Scholar
Matuszewska, W. (1962), ‘Regularly increasing functions in connection with the theory of L*ϕ-spaces’, Studia Math. 21, 317–344.CrossRefGoogle Scholar
Neveu, J. (1976), Processus ponctuels, Ecole d'Eté de Probabilités de Saint-Flour (Lecture Notes in Mathematics 598, Springer, Berlin).Google Scholar
Resnick, S. and Greenwood, P. (1979), ‘A bivariate stable characterization and domains of attraction’, J. Multivariate Anal. 9, 206–221.CrossRefGoogle Scholar
Simons, G. and Stout, W. (1978), ‘A weak invariance principle with applications to domains of attraction’, Ann. Probab. 6, 294–315.CrossRefGoogle Scholar
Whitt, W. (1980), ‘Some useful functions for functional limit theorems’, Math. Oper. Res. 5, 67–85.CrossRefGoogle Scholar