Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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SOME RESULTS ON CONDITIONALLY UNIFORMLY STRONG MIXING SEQUENCES OF RANDOM VARIABLES

  • Yuan, De-Mei (School of Mathematics and Statistics Chongqing Technology and Business University) ;
  • Hu, Xue-Mei (School of Mathematics and Statistics Chongqing Technology and Business University) ;
  • Tao, Bao (School of Mathematics and Statistics Chongqing Technology and Business University)
  • Received : 2013.11.05
  • Published : 2014.05.01
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Abstract

From the ordinary notion of uniformly strong mixing for a sequence of random variables, a new concept called conditionally uniformly strong mixing is proposed and the relation between uniformly strong mixing and conditionally uniformly strong mixing is answered by examples, that is, uniformly strong mixing neither implies nor is implied by conditionally uniformly strong mixing. A couple of equivalent definitions and some of basic properties of conditionally uniformly strong mixing random variables are derived, and several conditional covariance inequalities are obtained. By means of these properties and conditional covariance inequalities, a conditional central limit theorem stated in terms of conditional characteristic functions is established, which is a conditional version of the earlier result under the non-conditional case.

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References

  1. P. Y. Chen and D. C. Wang, Complete moment convergence for sequence of identically distributed $\varphi$-mixing random variables, Acta Math. Sin. (Engl. Ser.) 26 (2010), no. 4, 679-690. https://doi.org/10.1007/s10114-010-7625-6
  2. Y. S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martingales, 3rd Edition, New York, Springer-Verlag, 1997.
  3. R. L. Dobrushin, Central limit theorem for nonstationary Markov chains. I, Theory Probab. Appl. 1 (1956), no. 1, 65-80. https://doi.org/10.1137/1101006
  4. R. L. Dobrushin, Central limit theorem for nonstationary Markov chains. II, Theory Probab. Appl. 1 (1956), no. 4, 329-383. https://doi.org/10.1137/1101029
  5. J. L. Doob, Stochastic Processes, New York, John Wiley and Sons, 1953.
  6. S. Hu and X. Wang, Large deviations for some dependent sequences, Acta Math. Sci. Ser. B Engl. Ed. 28 (2008), no. 2, 295-300.
  7. I. A. Ibragimov, Some limit theorems for stochastic processes stationary in the strict sense, Dokl. Akad. Nayk SSSR. 125 (1959), 711-714 (in Russian).
  8. I. A. Ibragimov, Some limit theorems for stationary processes, Theory Probab. Appl. 7 (1962), no. 4, 349-382. https://doi.org/10.1137/1107036
  9. K. Knopp, Theory and Application of Infinite Series, London, Blackie & Son, 1951.
  10. A. Kuczmaszewska, On the strong law of large numbers for $\varphi$-mixing and $\rho$-mixing random variables, Acta Math. Hungar. 132 (2011), no. 1-2, 174-189. https://doi.org/10.1007/s10474-011-0089-z
  11. Z. Y. Lin and Z. D. Bai, Probability Inequalities, Beijing, Science Press, 2010.
  12. M. Ordonez Cabrera, A. Rosalsky, and A. Volodin, Some theorems on conditional mean convergence and conditional almost sure convergence for randomly weighted sums of dependent random variables, TEST 21 (2012), no. 2, 369-385. https://doi.org/10.1007/s11749-011-0248-0
  13. M. Peligrad, An invariance principle for $\varphi$-mixing sequences, Ann. Probab. 13 (1985), no. 4, 1304-1313. https://doi.org/10.1214/aop/1176992814
  14. B. L. S. Prakasa Rao, Conditional independence, conditional mixing and conditional association, Ann. Inst. Statist. Math. 61 (2009), no. 2, 441-460. https://doi.org/10.1007/s10463-007-0152-2
  15. G. G. Roussas and A. Ioannides, Moment inequalities for mixing sequences of random variables, Stoch. Anal. Appl. 5 (1987), no. 1, 61-120.
  16. P. K. Sen, A note on weak convergence of empirical processes for sequences of $\varphi$-mixing random variables, Ann. Math. Statist. 42 (1971), no. 6, 2131-2133. https://doi.org/10.1214/aoms/1177693079
  17. Q. M. Shao, Almost sure invariance principles for mixing sequences of random variables, Stochastic Proc. Appl. 48 (1993), no. 2, 319-334. https://doi.org/10.1016/0304-4149(93)90051-5
  18. Z. S. Szewczak, A note on Marcinkiewicz laws for strictly stationary $\varphi$-mixing sequences, Statist. Probab. Lett. 81 (2011), no. 11, 1738-1741. https://doi.org/10.1016/j.spl.2011.06.001
  19. S. A. Utev, The central limit theorem for $\varphi$-mixing arrays of random variables, Theory Probab. Appl. 35 (1990), no. 1, 131-139.
  20. D. M. Yuan, J. An, and X. S.Wu, Conditional limit theorems for conditionally negatively associated random variables, Monatsh. Math. 161 (2010), no. 4, 449-473. https://doi.org/10.1007/s00605-010-0196-x
  21. D. M. Yuan and L. Lei, Some conditional results for conditionally strong mixing se-quences of random variables, Sci. China Math. 56 (2013), no. 4, 845-859. https://doi.org/10.1007/s11425-012-4554-0
  22. D. M. Yuan, L. R. Wei, and L. Lei, Conditional central limit theorems for a sequence of conditionally independent random variables, J. Korean Math. Soc. 51 (2014), no. 1, 1-15.
  23. D. M. Yuan and Y. Xie, Conditional limit theorems for conditionally linearly negative quadrant dependent random variables, Monatsh. Math. 166 (2012), no. 2, 281-299. https://doi.org/10.1007/s00605-012-0373-1
  24. D. M. Yuan and Y. K. Yang, Conditional versions of limit theorems for conditionally associated random variables, J. Math. Anal. Appl. 376 (2011), no. 1, 282-293. https://doi.org/10.1016/j.jmaa.2010.10.046
  25. D. M. Yuan and J. H. Zheng, Conditionally negative association resulting from multi-nomial distribution, Statist. Probab. Lett. 83 (2013), no. 10, 2222-2227. https://doi.org/10.1016/j.spl.2013.06.004

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