The strong law of large numbers for extended negatively dependent random variables. (English) Zbl 1213.60058

A sequence of \(\{X_k, k=1,2,\dots\}\) of random variables is said to be extended negatively dependent if for each n the tails of finite-dimensional distributions of random variables \(\{X_k, k=1,2,\dots,n\}\) in the lower-left and the upper-right corners are dominated by multiple of tails of corresponding distributions of a sequence of independent random variables with the same marginal distributions as have the random variables \(X_k\). If such statement holds only for multi-dimensional distributions in the lower-left corner only then the sequence \(\{X_k, k=1,2,\dots\}\) is called lower extended negative dependent (LEND), if it holds for the upper-right corner only, then we deal with the upper extended negative dependent (UEND) sequence. Sufficient conditions for LEND or for UEND are given in Lemma 2.1. By this lemma every \(n\)-dimensional Farlier-Gumber-Morgenstern distribution describes a specifies END structure.
The references on investigations in cases of various negative dependences are given in the Introduction.
The main statement is Theorem 1.1. Let \(\{X_k, k=1,2,\dots\}\) be a sequence of END random variables with common distribution \(F\). Let \(S_n\) be its nth partial sum, \(n=1,2,\dots\), then \(\frac{S_n}{n}\rightarrow \mu\) as \(n\rightarrow \infty\) for some real number \(\mu\) if and only if \(E|X_1|<\infty\) and \(\mu = EX_1\).
In Section 2 six lemmas are presented five of which need proving and one of the is new even for the independent case. Theorem 1.1. is proved in section 3. Section 4 contains two applications of Theorem 1.1. to risk theory and renewal theory.
The volume of the paper is 15 pages. The list of references contains 25 positions.


60F15 Strong limit theorems
60K05 Renewal theory
Full Text: DOI


[1] Aleškevičien\.e, A., Lepus, R. and Šiaulys, J. (2008). Tail behavior of random sums under consistent variation with applications to the compound renewal risk model. Extremes 11, 261-279. · Zbl 1164.60006
[2] Asmussen, S. (2003). Applied Probability and Queues , 2nd edn. Springer, New York. · Zbl 1029.60001
[3] Baek, J.-I., Seo, H.-Y., Lee, G.-H. and Choi, J.-L. (2009). On the strong law of large numbers for weighted sums of arrays of rowwise negatively dependent random variables. J. Korean Math. Soc. 46, 827-840. · Zbl 1175.60024
[4] Bingham, N. H. and Nili Sani, H. R. (2004). Summability methods and negatively associated random variables. In Stochastic Methods and Their Applications (J. Appl. Prob. 41A ), eds J. Gani and E. Seneta, Applied Probability Trust, Sheffield, pp. 231-238. · Zbl 1049.60024
[5] Block, H. W., Savits, T. H. and Shaked, M. (1982). Some concepts of negative dependence. Ann. Prob. 10, 765-772. · Zbl 0501.62037
[6] Chen, Y., Yuen, K. C. and Ng, K. W. (2010). Precise large deviations of random sums in presence of negative dependence and consistent variation. To appear in Methodology Comput. Appl. Prob. . · Zbl 1242.60027
[7] Cossette, H., Marceau, E. and Marri, F. (2008). On the compound Poisson risk model with dependence based on a generalized Farlie-Gumbel-Morgenstern copula. Insurance Math. Econom. 43, 444-455. · Zbl 1151.91565
[8] Denisov, D., Foss, S. and Korshunov, D. (2010). Asymptotics of randomly stopped sums in the presence of heavy tails. Bernoulli 16 , 971-994. · Zbl 1208.60041
[9] Ebrahimi, N. and Ghosh, M. (1981). Multivariate negative dependence. To appear in Commun. Statist. Theory Meth. 10, 307-337. · Zbl 0506.62034
[10] Gerasimov, M. Yu. (2009). The strong law of large numbers for pairwise negatively dependent random variables. Moscow Univ. Comput. Math. Cybernet. 33, 51-58. · Zbl 1178.60025
[11] Hashorva, E. (2001). Asymptotic results for FGM random sequences. Statist. Prob. Lett. 54, 417-425. · Zbl 0996.60039
[12] Joe, H. (1997). Multivariate Models and Dependence Concepts . Chapman and Hall, London. · Zbl 0990.62517
[13] Ko, B. and Tang, Q. (2008). Sums of dependent nonnegative random variables with subexponential tails. J. Appl. Prob. 45, 85-94. · Zbl 1137.62310
[14] Kočetova, J., Leipus, R. and Šiaulys, J. (2009). A property of the renewal counting process with application to the finite-time ruin probability. Lithuanian Math. J. 49, 55-61. · Zbl 1185.60098
[15] Kochen, S. and Stone, C. (1964). A note on the Borel-Cantelli lemma. Illinois J. Math. 8, 248-251. · Zbl 0139.35401
[16] Kotz, S., Balakrishnan, N. and Johnson, N. L. (2000). Continuous Multivariate Distributions , Vol. 1, 2nd edn. Wiley-Interscience, New York. · Zbl 0946.62001
[17] Liu, L. (2009). Precise large deviations for dependent random variables with heavy tails. Statist. Prob. Lett. 79, 1290-1298. · Zbl 1163.60012
[18] Matuła, P. (1992). A note on the almost sure convergence of sums of negatively dependent random variables. Statist. Prob. Lett. 15, 209-213. · Zbl 0925.60024
[19] Nelsen, R. B. (2006). An Introduction to Copulas , 2nd edn. Springer, New York. · Zbl 1152.62030
[20] Robert, C. Y. and Segers, J. (2008). Tails of random sums of a heavy-tailed number of light-tailed terms. Insurance Math. Econom. 43, 85-92. · Zbl 1154.60032
[21] Tang, Q. (2006). Insensitivity to negative dependence of the asymptotic behavior of precise large deviations. Electron. J. Prob. 11, 107-120. · Zbl 1109.60021
[22] Tang, Q. and Tsitsiashvili, G. (2003). Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stoch. Process. Appl. 108, 299-325. · Zbl 1075.91563
[23] Tang, Q. and Vernic, R. (2007). The impact on ruin probabilities of the association structure among financial risks. Statist. Prob. Lett. 77, 1522-1525. · Zbl 1210.91061
[24] Tang, Q. and Yan, J. (2002). A sharp inequality for the tail probabilities of sums of i.i.d. r.v.’s with dominatedly varying tails. Sci. China Ser. A 45, 1006-1011. · Zbl 1099.60501
[25] Yan, J. (2006). A simple proof of two generalized Borel-Cantelli lemmas. In Memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX (Lecture Notes Math. 1874 ), Springer, Berlin, pp. 77-79. · Zbl 1118.60005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.