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Convolutions of Distributions With Exponential and Subexponential Tails

Published online by Cambridge University Press:  09 April 2009

Daren B. H. Cline
Affiliation:
Department of StatisticsTexas A & M UniversityCollege Station, Texas 77843, U.S.A.
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Distribution tails F(t) = F(t, ∞) are considered for which and as t → ∞. A real analytic proof is obtained of a theorem by Chover, Wainger and Ney, namely that .

In doing so, a technique is introduced which provides many other results with a minimum of analysis. One such result strengthens and generalizes the various known results on distribution tails of random sums.

Additionally, the closure and factorization properties for subexponential distributions are investigated further and extended to distributions with exponential tails.

MSC classification

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1987

References

Chistyakov, V. P. (1964), ‘A theorem on sums of independent positive random variables and its applications to branching random processes,’ Theory Probab. Appl. 9, 640648.CrossRefGoogle Scholar
Chover, J., Ney, P. and Wainger, S. (1973a), ‘Functions of probability measures,’ J. Analyse Math. 26, 255302.CrossRefGoogle Scholar
Chover, J., Ney, P. and Wainger, S. (1973b), ‘Degeneracy properties of subcritical branching process,’ Ann Probab. 1, 663673.CrossRefGoogle Scholar
Cline, D. B. H. (1986), ‘Convolution tails, product tails and domains of attraction’, Probab. Theor. Rel. Fields 72, 525557.CrossRefGoogle Scholar
Davis, R. A. and Resnick, S. I. (1985a), ‘Limit theory for moving averages of random variables with regularly varying tail probabilities,‘ Ann. Probab. 13, 179195.CrossRefGoogle Scholar
Davis, R. A. and Resnick, S. I. (1985b), ‘More limit theory for the sample correlation function of moving averages’, Stochastic Process. Appl. 20, 257279.CrossRefGoogle Scholar
Embrechts, P. (1983), ‘The asympotic behaviour of series and power series with positive coefficients’, Verh. Konink. Acad. Wetensch., Belgie.Google Scholar
Embrechts, P. and Goldie, C. M. (1980), ‘On closure and factorization properties of subexponential distributions’, J. Aust. Math. Soc. (Ser. A) 29, 243256.CrossRefGoogle Scholar
Embrechts, P. and Goldie, C. M. (1982), ‘On convolution tails,’ Stochastic Process. Appl. 13, 263278.CrossRefGoogle Scholar
Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979), ‘Subexponentially and infinite divisibility’, Z. Wahrsch. Verw. Gebiete 49, 335347.CrossRefGoogle Scholar
Feller, W. (1971), An introduction to probability theory and its applications, Vol. II (Wiley).Google Scholar
Rudin, W. (1973), ‘Limits of ratios of tails of measures’, Ann. Probab. 1, 982994.CrossRefGoogle Scholar
Seneta, E. (1976), Regularly varying functions (Lecture Notes in Mathematics 508, Springer-Verlag).CrossRefGoogle Scholar