Precise large deviations for widely orthant dependent random variables with dominatedly varying tails

Abstract

For the widely orthant dependent (WOD) structure, this paper mainly investigates the precise large deviations for the partial sums ofWOD and non-identically distributed random variables with dominatedly varying tails. The obtained results extend some corresponding results.

Keywords

Precise large deviations / widely orthant dependent (WOD) / dominatedly varying tails

References

[1] Bingham N. H., Goldie C. M., Teugels J. L. Regular Variation, 1987, Cambridge: Cambridge University Press

[2] Block H. W., Savits T. H., Shaked M. Some concepts of negative dependence. Ann Probab, 1982, 10: 765-772

[3] Chen Y., Chen A., Ng K. W. The strong law of large numbers for extended negatively dependent random variables. J Appl Prob, 2010, 47: 908-922

[4] Chen Y., Yuen K. C., Ng K. W. Precise large deviations of random sums in presence of negative dependence and consistent variation. Methodol Comput Appl Probab, 2011, 13: 821-833

[5] Cline D. B. H. Intermediate regular and ∏ variation. Proc Lond Math Soc, 1994, 68: 594-616

[6] Cline D B H, Hsing T. Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails. Texas A&M University, Preprint, 1991

[7] Ebrahimi N., Ghosh M. Multivariate negative dependence. Comm Statist Theory Methods, 1981, 10: 307-337

[8] Embrechts P., Klüppelberg C., Mikosch T. Modelling Extremal Events for Insurance and Finance, 1997, Berlin: Springer

[9] Embrechts P., Omey E. A property of long-tailed distribution. J Appl Probab, 1984, 21: 80-87

[10] Heyde C. C. A contribution to the theory of large deviations for sums of independent random variables. Z Wahrscheinlichkeitsth, 1967, 7: 303-308

[11] Heyde C. C. On large deviation problems for sums of random variables which are not attracted to the normal law. Ann Math Statist, 1967, 38: 1575-1578

[12] Heyde C. C. On large deviation probabilities in the case of attraction to a non-normal stable law. Sankhyā, 1968, 30: 253-258

[13] Joag-Dev K., Proschan F. Negative association of random variables with applications. Ann Statist, 1983, 11: 286-295

[14] Liu L. Precise large deviations for dependent random variables with heavy tails. Statist Probab Lett, 2009, 79: 1290-1298

[15] Liu Y. Precise large deviations for negatively associated random variables with consistently varying tails. Statist Probab Lett, 2007, 77: 181-189

[16] Mikosch T., Nagaev A. V. Large deviations of heavy-tailed sums with applications in insurance. Extremes, 1998, 1: 81-110

[17] Nagaev A. V. Integral limit theorems for large deviations when Cramer’s condition is not fulfilled I. Theory Probab Appl, 1969, 14: 51-64

[18] Nagaev A. V. Integral limit theorems for large deviations when Cramer’s condition is not fulfilled II. Theory Probab Appl, 1969, 14: 193-208

[19] Nagaev S. V. Large deviations of sums of independent random variables. Ann Probab, 1979, 7: 754-789

[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, 2004, 41: 93-107

[21] Tang Q. Insensitivity to negative dependence of the asymptotic behavior of precise deviations. Electron J Probab, 2006, 11: 107-120

[22] 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, 2003, 108: 299-325

[23] Wang K, Wang Y, Gao Q. Uniform asymptotics for the finite-time ruin probability of a new dependent risk model with a constant interest rate. Methodol Comput Appl Probab, 2010, DOI: 10.1007/s11009-011-9226-y

[24] Wang Y., Wang K., Cheng D. Precise large deviations for sums of negatively associated random variables with common dominatedly varying tails. Acta Math Sin (Engl Ser), 2006, 22: 1725-1734

[25] Yang Y, Wang K. Precise large deviations for dependent random variables with applications to the compound renewal risk model. Rocky Mountain J Math (to appear)