Journal of Applied Probability

The strong law of large numbers for extended negatively dependent random variables

Yiqing Chen, Anyue Chen, and Kai W. Ng

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A sequence of random variables is said to be extended negatively dependent (END) if the tails of its finite-dimensional distributions in the lower-left and upper-right corners are dominated by a multiple of the tails of the corresponding finite-dimensional distributions of a sequence of independent random variables with the same marginal distributions. The goal of this paper is to establish the strong law of large numbers for a sequence of END and identically distributed random variables. In doing so we derive some new inequalities of large deviation type for the sums of END and identically distributed random variables being suitably truncated. We also show applications of our main result to risk theory and renewal theory.

Article information

J. Appl. Probab. Volume 47, Number 4 (2010), 908-922.

First available in Project Euclid: 4 January 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F15: Strong theorems
Secondary: 60K05: Renewal theory

Asymptotics Borel-Cantelli lemma lower/upper extended negative dependence renewal counting process strong law of large numbers truncation


Chen, Yiqing; Chen, Anyue; Ng, Kai W. The strong law of large numbers for extended negatively dependent random variables. J. Appl. Probab. 47 (2010), no. 4, 908--922. doi:10.1239/jap/1294170508.

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  • 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.
  • Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.
  • 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.
  • 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.
  • Block, H. W., Savits, T. H. and Shaked, M. (1982). Some concepts of negative dependence. Ann. Prob. 10, 765–772.
  • 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..
  • 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.
  • Denisov, D., Foss, S. and Korshunov, D. (2010). Asymptotics of randomly stopped sums in the presence of heavy tails. Bernoulli 16, 971–994.
  • Ebrahimi, N. and Ghosh, M. (1981). Multivariate negative dependence. To appear in Commun. Statist. Theory Meth. 10, 307–337.
  • Gerasimov, M. Yu. (2009). The strong law of large numbers for pairwise negatively dependent random variables. Moscow Univ. Comput. Math. Cybernet. 33, 51–58.
  • Hashorva, E. (2001). Asymptotic results for FGM random sequences. Statist. Prob. Lett. 54, 417–425.
  • Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.
  • Ko, B. and Tang, Q. (2008). Sums of dependent nonnegative random variables with subexponential tails. J. Appl. Prob. 45, 85–94.
  • 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.
  • Kochen, S. and Stone, C. (1964). A note on the Borel–Cantelli lemma. Illinois J. Math. 8, 248–251.
  • Kotz, S., Balakrishnan, N. and Johnson, N. L. (2000). Continuous Multivariate Distributions, Vol. 1, 2nd edn. Wiley-Interscience, New York.
  • Liu, L. (2009). Precise large deviations for dependent random variables with heavy tails. Statist. Prob. Lett. 79, 1290–1298.
  • Matuła, P. (1992). A note on the almost sure convergence of sums of negatively dependent random variables. Statist. Prob. Lett. 15, 209–213.
  • Nelsen, R. B. (2006). An Introduction to Copulas, 2nd edn. Springer, New York.
  • 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.
  • Tang, Q. (2006). Insensitivity to negative dependence of the asymptotic behavior of precise large deviations. Electron. J. Prob. 11, 107–120.
  • 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.
  • Tang, Q. and Vernic, R. (2007). The impact on ruin probabilities of the association structure among financial risks. Statist. Prob. Lett. 77, 1522–1525.
  • 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.
  • 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.