A class of dependent random variables‎, ‎properties and applications

Document Type : Original Paper


Department of Statstics, University of Birjand, Birjand , Iran.


‎In this paper‎, ‎after calling a class of dependent random variables, APND, that contains some big classes of negatively dependent and some classes of positively dependent random variables‎, ‎the relationship of this class of random variables with well known classes of dependent variables are explained and some basic relations‎, ‎contain moment and maximal inequalities are proofed‎. ‎At the end‎, ‎the limiting behavior of an arbitrary array of random variables is studied‎.


Main Subjects

[1] N.H. Bingham and H.R. Nili Sani, Summability methods and negatively associated random variables,
Journal of Applied Probability. 41(A) (2004) 231-238.
[2] A. Bozorgnia and R.F. Patterson and R.L. Taylor, Limit theorems for dependent random variables,
Proceedings of the first world congress on World congress of nonlinear analysts, 92 volume II (1996)
[3] Q. Dehua, and K.C. Chang and R.G. Antonini and A. Volodin, On the strong rates of convergence for
arrays of rowwise negatively dependent random variables, Stochastic Analysis and Applications,29
(2011) 375-385.
[4] C. Genest and B. Remillard and D. Beaudoinc, Goodness-of-fit tests for copulas: A review and a
power study, Insurance: Mathematics and Economic, 44 (2009) 199–213.
[5] S. Kotz and Y. Lumelskii and M. Pensky, The stress-strength model and its generalizations-Theory
and Applications,World Scientific Publishing Co., 2003.
[6] H. Joe, Dependence modeling with copulas, CRC press, 2015.
[7] H.S. Konijn, Positive and negative depencence two random variables, Sankhyā: The Indian Journal
of Statistic, 2 (1959) 269-280.
[8] E.L. Lehmann, Some concepts of dependence, Ann. Math. Stat., 31 (1966) 1137–1153.
[9] D.D. Mari and S. Kotz, Correlation and Dependene, Imperial College Press, 2004.
[10] R.B. Nelson, An Introduction to Copulas, Springer Science & Business Media, 2006.
[11] H.R. Nili Sani and M. Amini and A. Bozorgnia, Strong Laws for Weighted Sums of Negative dependent
Random Variables, J. of Sci., I.R.I, 16(3) (2005) 261–265.
[12] H.R. Nili Sani and M. Amini and A. Bozorgnia, Complete convergence for weighted sums of arrays
of APND random variables, Commun. Stat.Theory Methods, 7(10) (2018) 2425-2431.
[13] V.V. Petrov, Limit theorems of probability theory, Clarendon Press, 1995.
[14] D. Qiu and P. Chen and R.G. Antonini and A. Volodin, On the complete convergence for arrays of
rowwise extended negatively dependent random variables, J. Korean Math. Soc., 2 (2013) 379–392.
[15] V. Ranjbar and M. Amini and A. Bozorgnia, Asymptoti Behavior Weighted Sums of Weakly Negative
dependent Random Variables, Journal of Sienes Islami Republi of Iran, 19 (2008) 357-363.
[16] B.P. Rao, Associated sequences, demimartingales and nonparametric inference, Springer Science &
Business Media, 2012.
[17] Q.M. Shao, A comparison theorem on moment inequalities between negatively associated and independent
random variables, Journal of Theoretical Probability ,13(2) (2000) 343-356.
[18] G. Shixin, On On almost sure convergence of weighted sums of random element sequences, Acta
Mathematica Scientia, 30B(4) (2010) 1021-1028.
[19] A. Sklar, Fonctions de repartition an dimensions et leurs marges. Publ. Inst. Statist. Univ.
Paris,8(1959) 229–231.
[20] K. Wang and Y. Wang and Gao and Q. Qiu, Uniform asymptotics for the finite-time ruin probability
of a dependent risk model with a constant interest rate, Methodol Comput. Appl. Proba., 15 (2013)
[21] Y. Yi and D. Qiu, A note on the Kolmogorov-Feller weak law of large numbers, Journal of Mathematical
Research with Applications, 35(2) (2015) 223–228.
[22] K. Zhang and J. Lin and C. HUANG, Some new results on weighted geometric mean for copulas,
International Journal of Uncertainty Fuzziness and Knowledge-Based Systems , 21(2) (2013) 277–288.