Abstract
We consider real-valued random variables X 1,…,X n with corresponding distributions F 1,…, F n such that X 1,…,X n admit some dependence structure and n −1(F 1 +· · ·+F n ) belongs to the class of dominatedly varyingtailed distributions. We establish weak equivalence relations among P(S n > x), P(max{X 1,…,X n } > x), P(max{S 1,…,S n } > x), and \( {\displaystyle {\sum}_{k=1}^n\overline{F_k}(x)} \) as x → ∞, where S k := X 1 + · · · + X k . Some copula-based examples illustrate the results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Amblard and S. Girard, A new extension of bivariate FGM copulas, Metrika, 70:1–17, 2009.
I. Bairamov, S. Kotz, and M. Bekçi, New generalized Farlie–Gumbel–Morgenstern distributions and concomitants of order statistics, J. Appl. Stat., 28:521–536, 2001.
H. Bekrizadeh, G.A. Parham, and M.R. Zadkarmi, A new generalization of Farlie–Gumbel–Morgenstern copulas, Appl. Math. Sci., 6:3527–3533, 2012.
H.W. Block, T.H. Savits, and M. Shaked, Some concepts of negative dependence, Ann. Probab., 10:765–772, 1982.
Y. Chen and K.C. Yuen, Sums of pairwise quasi-asymptotically independent random variables with consistent variation, Stoch. Models, 25:76–89, 2009.
N. Ebrahimi andM. Ghosh, Multivariate negative dependence, Commun. Stat., Theory Methods, 10:307–337, 1981.
D.J.G. Farlie, The performance of some correlation coefficients for a general bivariate distribution, Biometrika, 47:307–323, 1960.
J. Geluk and Q. Tang, Asymptotic tail probabilities of sums of dependent subexponential random variables, J. Theor. Probab., 22:871–882, 2009.
E.J. Gumbel, Bivariate exponential distributions, J. Am. Stat. Assoc., 55:698–707, 1960.
J.S. Huang and S. Kotz, Modifications of the Farlie–Gumbel–Morgenstern distributions. A tough hill to climb, Metrika, 49:135–145, 1999.
T. Jiang, Q. Gao, and Y. Wang, Max-sum equivalence of conditionally dependent random variables, Stat. Probab. Lett., 84:60–66, 2014.
J. Li and Q. Tang, A note on max-sum equivalence, Stat. Probab. Lett., 80:1720–1723, 2010.
L. Liu, Precise large deviations for dependent random variables with heavy tails, Stat. Probab. Lett., 79:1290–1298, 2009.
X. Liu, Q. Gao, and Y. Wang, A note on a dependent risk model with constant interest rate, Stat. Probab. Lett., 82:707–712, 2012.
D.S. Mitrinović, Analytic Inequalities, Springer, New York, 1970.
D. Morgenstern, Einfache Beispiele zweidimensionaler Verteilungen, Mitt.-Bl. Math. Statistik, 8:234–235, 1956.
Q. Tang, Insensitivity to negative dependence of the asymptotic behavior of precise large deviations, Electron. J. Probab., 11:107–120, 2006.
Y. Yang, R. Leipus, and L. Dindienė, On the max-sum equivalence in presence of negative dependence and heavy tails, Information Technology and Control, 44(2):215–220, 2015.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author is supported by a grant (No. MIP-13079) from the Research Council of Lithuania.
Rights and permissions
About this article
Cite this article
Dindienė, L., Leipus, R. Weak max-sum equivalence for dependent heavy-tailed random variables. Lith Math J 56, 49–59 (2016). https://doi.org/10.1007/s10986-016-9303-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10986-016-9303-6