Abstract
Let {X,X k : k ≥ 1} be a sequence of extended negatively dependent random variables with a common distribution F satisfying EX > 0. Let τ be a nonnegative integer-valued random variable, independent of {X,X k : k ≥ 1}. In this paper, the authors obtain the necessary and sufficient conditions for the random sums \(S_\tau = \sum\limits_{n = 1}^\tau {X_n } \) to have a consistently varying tail when the random number τ has a heavier tail than the summands, i.e.,
as x→∞.
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Aleškevičiené, A., Leipus, R. and Šiaulys, J., Tail behavior of random sums under consistent variation with applications to the compound renewal risk model, Extremes, 11, 2008, 261–279.
Bingham, N. H., Goldie, C. M. and Teugels, J. L., Regular Variation, Cambridge University Press, Cambridge, 1987.
Chen, Y. Q., Chen, A. Y. and Ng, K. W., The strong law of large numbers for extended negatively dependent random variables, J. Appl. Probab., 47, 2010, 908–922.
Cline, D. B. H., Convolutions of distributions with exponential and subexponential tails, J. Aust. Math. Soc. Ser. A, 43, 1987, 347–365.
Embrechts, P., Goldie, C. M. and Veraverbeke, N., Subexponentiality and infinite divisibility, Z. Wahrsch., 49, 1979, 335–347.
Embrechts, P., Klüppelberg, C. and Mikosch, T., Modelling Extremal Events for Insurance and Finance, Springer-Verlag, Berlin, 1997.
Faÿ, G., González-Arévalo, B., Mikosch, T. and Samorodnitsky, G., Modelling teletraffic arrivals by a Poisson cluster process, Queueing Syst. Theor. Appl., 54, 2006, 121–140.
Foss, S., Korshunov, D. and Zachary, S., Convolutions of long-tailed and subexponential distributions, J. Appl. Probab., 46, 2009, 756–767.
Liu, L., Precise large deviations for dependent random variables with heavy tails, Statist. Probab. Lett., 79, 2009, 1290–1298.
Pakes A., Convolution equivalence and infinite divisibility, J. Appl. Probab., 41, 2004, 407–424.
Robert, C. Y. and Segers, J., Tails of random sums of a heavy-tailed number of light-tailed terms, Insurance Math. Econom., 43, 2008, 85–92.
Tang, Q. H., Insensitivity to negative dependence of the asymptotic behavior of precise large deviations, Electron. J. Probab., 11, 2006, 107–120.
Watanabe, T., Convolution equivalence and distributions of random sums, Probab. Theory Related Fields, 142, 2008, 367–397.
Yu, C. J., Wang, Y. B. and Yang, Y., The closure of the convolution equivalent distribution class under convolution roots with applications to random sums, Statist. Probab. Lett., 80, 2010, 462–472.
Zhang, J., Cheng, F. Y. and Wang, Y. B., Tail behavior of random sums of negatively associated increments, J. Math. Anal. Appl., 376, 2011, 64–73.
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Project supported by the National Natural Science Foundation of China (No. 11071182).
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Cheng, F., Li, N. Asymptotics for the tail probability of random sums with a heavy-tailed random number and extended negatively dependent summands. Chin. Ann. Math. Ser. B 35, 69–78 (2014). https://doi.org/10.1007/s11401-013-0815-7
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DOI: https://doi.org/10.1007/s11401-013-0815-7