Abstract
We study the properties of subexponential distributions and find new sufficient and necessary conditions for membership in the class of these distributions. We establish a connection between the classes of subexponential and semiexponential distributions and give conditions for preservation of the asymptotics of subexponential distributions for “functions of distributions”. We address similar problems for the class of the so-called locally subexponential distributions. As an application of these results, we refine the asymptotics of the distribution of the supremum of sequential sums of random variables with negative drift, in particular, local theorems.
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References
Chistyakov V. P., “A theorem on sums of independent positive random variables and its application to branching random processes,” Theor. Probab. Appl., 9, 640-648 (1964).
Athreya K. B. and Ney P. E., Branching Processes, Springer-Verlag, Berlin (1975).
Chover J., Ney P. E., and Wainger S., “Functions of probability measures,” J. Anal. Math., 26, 255-302 (1973).
Chover J., Ney P. E., and Wainger S., “Degeneracy properties of subcritical branching processes,” Ann. Probab., 1, 663-673 (1974).
Teugels J. L., “The class of subexponential distributions,” Ann. Probab., 3, No. 6, 1000-1011 (1975).
Bingham N. H., Goldie C. M., and Teugels J. L., Regular Variation, Cambridge Univ. Press, Cambridge (1987).
Goldie C. M., “Subexponential distributions and dominated-variation tails,” J. Appl. Probab., 15, 440-442 (1978).
Klüppelberg C., “Subexponential distributions and integrated tails,” J. Appl. Probab., 25, No. 1, 132-141 (1988).
Pitman K. J. G., “Subexponential distribution functions,” J. Austral. Math. Soc., 29, 337-347 (1980).
Embrechts P., Goldie C. M., and Veraverbeke N., “Subexponentiality and infinite divisibility,” Z. Wahrsch. Verw. Gebiete, 49, 335-347 (1979).
Embrechts P. and Goldie C. M., “On convolution tails,” Stochastic Process. Appl., 13, 263-278 (1982).
Klüppelberg C., “Subexponential distributions and characterizations of related classes,” Probab. Theory Related Fields, 82, No. 2, 259-269 (1989).
Embrechts P., Klüppelberg C., and Mikosh T., Modelling Extremal Events for Insurance and Finance, Springer-Verlag, Berlin (1997).
Bertoin J. and Doney R. A., “On the local behaviour of ladder height distributions,” J. Appl. Probab., 31, 816-821 (1994).
Sgibnev M. S., “Banach algebras of functions with identical asymptotic behavior at infinity,” Sibirsk. Mat. Zh., 22, No. 3, 179-187 (1981).
Asmussen S., Kalashnikov V., Konstantinides D., Klüppelberg C., and Tsitsiashvili G., “A local limit theorem for random walk maxima with heavy tails,” Statist. Probab. Lett., 56, 399-404 (2002).
Borovkov A. A., Stochastic Processes in Queueing Theory [in Russian], Nauka, Moscow (1972).
Borovkov A. A., “Remarks on Wiener's and Blackwell's theorems,” Teor. Veroyatnost. i Primenen., 9, No. 2, 331-343 (1964).
Borovkov A. A., “On the asymptotics of distributions of the first passage times,” Mat. Zametki (to appear).
Veraverbeke N., “Asymptotic behaviour of Wiener-Hopf factors of a random walk,” Stochastic Process. Appl., 5, No. 1, 27-37 (1977).
Feller W., “On regular variation and local limit theorems,” in: Proc. Fifth Berkeley Sympos. Math. Statist. Probab. Vol. 2. Contributions to Probability Theory. Part I, Univ. of California Press, Berkeley, 1967, pp. 373-388.
Borovkov A. A. Probability [in Russian], 2 ed.: Nauka, Moscow (1986); 3 ed.: Éditorial, Moscow; Sobolev Inst. Mat., Novosibirsk (1999).
Borovkov A. A. and Borovkov K. A., “On large deviation probabilities for random walks. I: Regularly varying distribution tails,” Teor. Veroyatnost. i Primenen., 46, No. 2, 209-232 (2001).
Willekens E. and Teugels J. L., “Asymptotic expansions for waiting time probabilities in an M/G/1 queue with long-tailed service time,” Queueing Systems Theory Appl., 10, No. 4, 295-311 (1992).
Borovkov A. A., “Large deviation probabilities for random walks with semiexponential distributions,” Sibirsk. Mat. Zh., 41, No. 6, 1290-1324 (2000).
Borovkov A. A. and Boxma O. J., “On large deviation probabilities for random walks with heavy tails,” Siberian Adv. in Math. (to appear).
Borovkov A. A., “Estimates for the distribution of sums and maxima of sums of random variables without the Cramer condition,” Sibirsk. Mat. Zh., 41, No. 5, 997-1038 (2000).
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Borovkov, A.A. On Subexponential Distributions and Asymptotics of the Distribution of the Maximum of Sequential Sums. Siberian Mathematical Journal 43, 995–1022 (2002). https://doi.org/10.1023/A:1021109132124
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DOI: https://doi.org/10.1023/A:1021109132124