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Precise large deviations for dependent random variables with heavy tails. (English) Zbl 1163.60012

Summary: By extending the negatively dependent (ND) structure, the paper puts forth the concept of extended negative dependence (END). The results show that the END structure has no effect on the asymptotic behavior of precise large deviations of partial sums and random sums for non-identically distributed random variables on ( \(-\infty ,+\infty \)).

MSC:

60F10 Large deviations
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[1] Alam, K.; Saxena, K. M.L., Positive dependence in multivariate distributions, Commun. Stat. - Theory. Methods, 10, 1183-1196 (1981) · Zbl 0471.62045
[2] Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular Variation (1987), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0617.26001
[3] Block, H. W.; Savits, T. H.; Shaked, M., Some concepts of negative dependence, Ann. Probab., 10, 765-772 (1982) · Zbl 0501.62037
[4] Cline, D. B.H., Intermediate regular and \(\Pi\) variation, Proc. London Math. Soc., 68, 594-616 (1994) · Zbl 0793.26004
[5] Cline, D.B.H., Hsing, T., 1991. Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails, Texas A& M University, Preprint; Cline, D.B.H., Hsing, T., 1991. Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails, Texas A& M University, Preprint
[6] Cline, D. B.H.; Samorodnitsky, G., Subexponentiality of the product of independent random variables, Stochastic Process. Appl., 49, 75-98 (1994) · Zbl 0799.60015
[7] Embrechts, P.; Klüppelberg, C.; Mikosch, T., Modeling Extremal Events for Insurance and Finance (1997), Springer: Springer Berlin, Heidelberg · Zbl 0873.62116
[8] Heyde, C. C., On large deviation problems for sums of random variables which are not attracted to the normal law, Ann. Math. Statist., 38, 1575-1578 (1967) · Zbl 0189.51704
[9] Heyde, C. C., On large deviation probabilities in the case of attraction to a nonnormal stable law, Sankhyā, 30, 253-258 (1968) · Zbl 0182.22903
[10] Geluk, J.; Ng, K., Tail behavior of negatively associated heavy tailed sums, J. Appl. Probab., 43, 587-593 (2006) · Zbl 1104.60313
[11] Jelenković, P. R.; Lazar, A. A., Asymptotic results for multiplexing subexponential on-off processes, Adv. Appl. Probab., 31, 394-421 (1999) · Zbl 0952.60098
[12] Joag-Dev, K.; Proschan, F., Negative association of random variables with applications, Ann. Statist., 11, 286-295 (1983) · Zbl 0508.62041
[13] Kass, R.; Tang, Q., Note on the tail behavior of random walk maxima with heavy tails and negative drift, N. Am. Actrar. J., 7, 3, 57-61 (2003) · Zbl 1084.60515
[14] Ko, B.; Tang, Q., Sums of dependent nonnegative random variables with subexponential tails, J. Appl. Probab., 45, 85-94 (2008) · Zbl 1137.62310
[15] Lehmann, E. L., Some concepts of dependence, Ann. Math. Statist., 37, 1137-1153 (1966) · Zbl 0146.40601
[16] Meerschaert, M. M.; Scheffler, H. P., Limit Distributions for Sums of Independent Random Vectors. Heavy Tails in Theory and Practice (2001), Wiley: Wiley New York · Zbl 0990.60003
[17] Mikosch, T.; Nagaev, A. V., Large deviations of heavy-tailed sums with applications in insurance, Extremes, 1, 81-110 (1998) · Zbl 0927.60037
[18] Nagaev, A. V., Integral limit theorems for large deviations when Cramér’s condition is not fulfilled I, Theory Probab. Appl., 14, 51-64 (1969), II, 193-208 · Zbl 0196.21002
[19] Nagaev, S. V., Large deviations of sums of independent random variables, Ann. Probab., 7, 745-789 (1979) · Zbl 0418.60033
[20] Ng., K. W.; Tang, Q.; Yan, J.; Yang, H., Precise large deviations for sums of random variables with consistently varying tails, J. Appl. Probab., 41, 93-107 (2004) · Zbl 1051.60032
[21] Paulauskas, V.; Skučaitė, A., Some asymptotic results for one-sided large deviation probabilities, Lith. Math. J., 43, 3, 318-326 (2003) · Zbl 1048.60027
[22] Rozovski, L. V., Probabilities of large deviations on the whole axis, Theory Probab. Appl., 38, 53-79 (1993) · Zbl 0801.60021
[23] Schlegel, S., Ruin probabilities in perturbed risk models, Insurance Math. Econom., 22, 93-104 (1998) · Zbl 0907.90100
[24] Stadtmüller, U.; Trautner, R., Tauberian theorems for Laplace transforms. Tauberian theorems for Laplace transforms, J. Reine. Angew. Math., 311/312, 283-290 (1979) · Zbl 0409.44003
[25] Tang, Q., Insensitivity to negative dependence of the asymptotic behavior of precise large deviations, Electron. J. Probab., 11, 107-120 (2006) · Zbl 1109.60021
[26] Tang, Q.; Su, C.; Jiang, T.; Zhang, J. S., Large deviations for heavy-tailed random sums in compound renewal model, Statist. Probab. Lett., 52, 91-100 (2001) · Zbl 0977.60034
[27] Tang, Q.; Tsitsiashvili, G., Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks, Stochastic Process. Appl., 108, 299-325 (2003) · Zbl 1075.91563
[28] Vinogradov, V., Refined Large Deviation Limit Theorems (1994), Longman: Longman Harlow, Co-published in the United States with John Wiley & Sons, Inc., New York · Zbl 0832.60001
[29] Wang, D.; Tang, Q., Maxima of sums and random sums for negatively associated random variables with heavy tails, Statist. Probab. Lett., 68, 287-295 (2004) · Zbl 1116.62351
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