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Some Properties of stochastic compactness

Part of: Distribution theory - Probability Limit theorems Stochastic processes

Published online by Cambridge University Press:  09 April 2009

R. A. Maller
R. A. Maller
Affiliation:
CSIRO Division of Mathematics and Statistics, Perth, Western Australia
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Abstract

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The aim of this paper is to show that some of the known properties of distributions in the domain of attraction of a stable law have counterparts for distributions which are stochastically compact in the sense of Feller. This enables us to unify the ideas of Feller and Doeblin, who first studied the concept of stochastic compactness, and give new characterizations of stochastic compactness and the domain of attraction of the normal distribution.

MSC classification

Secondary: 60F05: Central limit and other weak theorems 60E07: Infinitely divisible distributions; stable distributions 60G50: Sums of independent random variables; random walks
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 30 , Issue 3 , February 1981 , pp. 264 - 277
Copyright
Copyright © Australian Mathematical Society 1981

References

de Haan, L. and Ridder, G. (1979), ‘Stochastic compactness of sample extremes’. Ann. Probability 7, 290–303.CrossRefGoogle Scholar
Doeblin, W. (1940), ‘Sur l'ensemble de puissances d'une loi de probabilité’, Studia. Math. 9, 71–96.CrossRefGoogle Scholar
Doeblin, W. (1947), ‘Sur l'ensemble de puissances d'une loi de probabilité’, Ann. Sci. Ecole Norm. 63, 317–350.CrossRefGoogle Scholar
Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979), ‘Subexponentiality and infinite divisiblity’, Z. Wahrsch. Verw. Gebiete 49, 335–347.CrossRefGoogle Scholar
Feller, W. (1965–66), ‘On regular variation and local limit theorems’. Proc. V Berkeley Symp. Math. Stats. Prob. II, part I, pp. 373–388.Google Scholar
Feller, W. (1969), ‘One sided analogues of Karamata's regular variation’. Enseignement Math. 15, 107–121.Google Scholar
Feller, W. (1971), An introduction to probability theory and its applications II (2nd ed., Wiley, New York).Google Scholar
Gnedenko, B. V. and Kolmogorov, A. N. (1968), Limit distributions for sums of independent random variables (2nd ed., Addison-Wesley, Reading, Mass.).Google Scholar
Goldie, C. M. (1977), ‘Convergence theorems for empirical Lorentz curves and their inverses’, Advances in Appl. Probability 9, 765–791.CrossRefGoogle Scholar
Goldie, C. M. (1978), ‘Subexponential dsitributions and dominated variation tails’, J. Appl. Probability 15, 440–442.CrossRefGoogle Scholar
Kesten, H. (1972), ‘Sums of independent random variables-without moment conditions’, Ann. Math. Statist. 43, 701–732.CrossRefGoogle Scholar
Letac, G. (1970), ‘On slow variation’, Proc. Amer. Math. Soc. 24, 745–746.CrossRefGoogle Scholar
Lévy, P. (1937), Théorie de l'addition des variables aléatoires (Gauthier-Villars, Paris).Google Scholar
Maller, R. A. (1978), ‘A local limit theorem for independent random variables’, Stochastic Processes Appl. 7, 101–111.CrossRefGoogle Scholar
Maller, R. A. (1980), ‘A note on domains of partial attraction’, Ann. Probability 8, 576–583.CrossRefGoogle Scholar
Maller, R. A. (1979), ‘Relative stability, stochastic compactness and characteristic functions’, J. Aust. Math. Soc. Ser. A 28, 499–509.CrossRefGoogle Scholar
Owen, W. L. (1973), ‘An estimate for E|S n| for variables in the domain of normal attraction of a stable law of index α, 1 < α < 2’, Ann. Probability 1, 1071–1073.CrossRefGoogle Scholar
Seneta, E. (1976), Regularly varying functions (Lecture Notes in Mathematics 508, Springer-Verlag, Berlin).CrossRefGoogle Scholar
Siegel, V. G. (1978), ‘Zero-one laws and weak convergences for sums of independent random variables’, Math. Nachr. 86, 333–346.CrossRefGoogle Scholar
Simons, G. and Stout, W. (1978), ‘A weak invariance principle with applications to domains of attraction’, Ann. Probability 6, 294–315.CrossRefGoogle Scholar
Smythe, R. T. (1974), ‘Sums of independent random variables on partially ordered sets’, Ann. Probability 2, 906–917.CrossRefGoogle Scholar
Thompson, M. E., Basu, A. K. and Owen, W. L. (1971), ‘On the existence of the optimal stopping rule in the Sn/n problem when the second moment is infinite’, Ann. Math. Statist. 42, 1936–1942.CrossRefGoogle Scholar
Thompson, M. E. and Owen, W. L. (1972), ‘The structure of the optimal stopping rule in the Sn/n problem’, Ann. Math. Statist. 43, 1110–1121.CrossRefGoogle Scholar