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Summability methods and negatively associated random variables

Part of: Limit theorems

Published online by Cambridge University Press:  14 July 2016

N. H. Bingham  and
H. R. Nili Sani
N. H. Bingham
Affiliation:
Department of Probability and Statistics, University of Sheffield, Sheffield S3 7RH, UK. Email address: nick.bingham@sheffield.ac.uk
H. R. Nili Sani
Affiliation:
Department of Mathematics, Birjand University, Birjand, Iran. Email address: nili@math.um.ac.ir
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Abstract

The paper studies convergence of sequences of negatively associated random variables under various summability methods. The results extend previously known results for independence and complement known results for ϕ-mixing.

Keywords

Negatively dependent random variablenegatively associated random variablesummability methods

MSC classification

Primary: 60F15: Strong theorems
Type
Part 5. Properties of random variables
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004 , pp. 231 - 238
Copyright
Copyright © Applied Probability Trust 2004 

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References

Bingham, N. H. (1984a). On Valiron and circle convergence. Math. Z. 186, 273–286.Google Scholar
Bingham, N. H. (1984b). Tauberian theorems for summability methods of random-walk type. J. London Math. Soc. 30, 281–287.Google Scholar
Bingham, N. H. (1986a). Extensions of the strong law. In Analytic and Geometric Stochastics (Adv. Appl. Prob. Suppl.), ed. Kendall, D. G., Applied Probability Trust, Sheffield, pp. 27–36.Google Scholar
Bingham, N. H. (1986b). Summability methods and dependent strong laws. In Dependence in Probability and Statistics (Progr. Prob. Statist. 11), Birkhäuser, Boston, MA, pp. 291–300.CrossRefGoogle Scholar
Bingham, N. H. (1988). Tauberian theorems for Jakimovski and Karamata-Stirling methods. Mathematika 35, 216–224.Google Scholar
Bingham, N. H. (1989). Moving averages. In Almost Everywhere Convergence , Academic Press, Boston, MA, pp. 131–144.Google Scholar
Bingham, N. H. and Goldie, C. M. (1988). Riesz means and self-neglecting functions. Math. Z. 199, 443–454.CrossRefGoogle Scholar
Bingham, N. H. and Maejima, M. (1985). Summability methods and almost sure convergence. Z. Wahrscheinlichkeitsth. 68, 383–392.CrossRefGoogle Scholar
Bingham, N. H. and Stadtmüller, U. (1990). Jakimovski methods and almost-sure convergence. In Disorder in Physical Systems , eds Grimmett, G. R. and Welsh, D. J. A., Oxford University Press, pp. 5–18.Google Scholar
Bingham, N. H. and Tenenbaum, G. (1986). Riesz and Valiron means and fractional moments. Math. Proc. Camb. Phil. Soc. 99, 143–149.Google Scholar
Block, H. W., Savits, T. H. and Shaked, M. (1985). A concept of negative dependence using stochastic ordering. Statist. Prob. Lett. 3, 81–86.Google Scholar
Bozorgnia, A., Patterson, R. F. and Taylor, R. L. (1996). Limit theorems for negatively dependent random variables. In Proc. 1st World Congr. Nonlinear Analysts (Tampa, FL, August 1992), Vol. 2, ed. Lakshmikantham, V., de Gruyter, Berlin, pp. 1639–1650.CrossRefGoogle Scholar
Chandra, T. K. and Ghosal, S. (1996a). Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables. Acta Math. Hungar. 71, 327–336.Google Scholar
Chandra, T. K. and Ghosal, S. (1996b). The strong law of large numbers for weighted average under dependence assumptions. J. Theoret. Prob. 9, 797–809.Google Scholar
Chandrasekharan, K. and Minakshisundaram, S. (1952). Typical Means. Oxford University Press.Google Scholar
Chow, Y. S. (1973). Delayed sums and Borel summability of independent identically distributed random variables. Bull. Inst. Math. Acad. Sinica 1, 207–220.Google Scholar
Chung, K.-L. (1974). A Course in Probability Theory , 2nd edn. Academic Press, New York.Google Scholar
Ebrahimi, N. and Ghosh, M. (1981). Multivariate negative dependence. Commun. Statist. Theory Meth. 10, 307–337.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications , Vol 2, 2nd edn. John Wiley, New York.Google Scholar
Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.Google Scholar
Hardy, G. H. (1949). Divergent Series. Clarendon Press, Oxford.Google Scholar
Ibragimov, I. A. and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.Google Scholar
Jakimovski, A. (1959). A generalization of the Lototosky method. Michigan Math. J. 6, 277–229.CrossRefGoogle Scholar
Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables with applications. Ann. Statist. 11, 286–295.CrossRefGoogle Scholar
Joe, H. (1997). Multivariate Models and Dependence Concepts (Monogr. Statist Appl. Prob. 73). Chapman and Hall, London.Google Scholar
Jogdeo, K. and Patil, G. P. (1975). Probability inequalities for certain multivariate discrete distributions. Sankhya B 37, 158–164.Google Scholar
Karlin, S. and Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities. J. Multivariate. Anal. 10, 499–516.Google Scholar
Kiesel, R. (1997). Strong laws and summability for sequences of ⊘-mixing random variables in Banach spaces. Electron. Commun. Prob. 2, 27–41.Google Scholar
Kiesel, R. (1998). Strong laws and summability for ⊘-mixing sequences of random variables. J. Theoret. Prob. 11, 209–234.CrossRefGoogle Scholar
Korevaar, J. (2004). A Century of Tauberian Theory. Springer, Berlin (to appear).Google Scholar
Lai, T. L. (1974). Summability methods for independent, identically distributed random variables. Proc. Amer. Math. Soc. 45, 253–261.Google Scholar
Lehmann, E. L. (1966). Some concepts of dependence. Ann. Math. Statist. 43, 1137–1153.Google Scholar
Liang, H.-Y, Chen, Z.-J. and Su, C. (2002). Convergence of Jamison-type weighted sums of pairwise negatively quadrant dependent random variables. Acta Math. Appl. Sinica 18, 161–168.Google Scholar
Matula, P. (1992). A note on the almost sure convergence of sums of negatively dependent random variables. Statist. Prob. Lett. 15, 209–213.CrossRefGoogle Scholar
Meir, A. (1963). On the [F, dn]-transformation of A. Jakimovski. Bull. Res. Council Israel F 10, 165–187.Google Scholar
Meyer-König, W. (1949). Untersuchungen über einige verwandte Limitierungsverfahren. Math. Z. 52, 257–304.Google Scholar
Newman, C. M. (1984). Asymptotic independence and limit theorems for positively and negatively dependent random variables. In Inequalities in Statistics and Probability (IMS Lecture Notes Monogr. Ser. 5), ed. Tong, Y. L., IMS, Hayward, CA, pp. 127–140,Google Scholar
Peligrad, M. (1985). Convergence rates of the strong law for stationary mixing sequences. Z. Wahrscheinlichkeitsth. 70, 307–314.CrossRefGoogle Scholar
Peligrad, M. (1989). The r-quick version of the strong law for stationary ⊘-mixing sequences. In Almost Everywhere Convergence , eds Edgar, G. A. and Sucheston, L., Academic Press, Boston, MA, pp. 335–348.Google Scholar
Shao, Q.-M. (2000). A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theoret. Prob. 13, 343–356.CrossRefGoogle Scholar
Shao, Q.-M. and Su, C. (1999). The law of the iterated logarithm for negatively associated random variables. Stoch. Process. Appl. 83, 139–148.Google Scholar