CONDITIONAL CENTRAL LIMIT THEOREMS FOR A SEQUENCE OF CONDITIONAL INDEPENDENT RANDOM VARIABLES

- Journal title : Journal of the Korean Mathematical Society
- Volume 51, Issue 1, 2014, pp.1-15
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2014.51.1.001

Title & Authors

CONDITIONAL CENTRAL LIMIT THEOREMS FOR A SEQUENCE OF CONDITIONAL INDEPENDENT RANDOM VARIABLES

Yuan, De-Mei; Wei, Li-Ran; Lei, Lan;

Yuan, De-Mei; Wei, Li-Ran; Lei, Lan;

Abstract

A conditional version of the classical central limit theorem is derived rigorously by using conditional characteristic functions, and a more general version of conditional central limit theorem for the case of conditionally independent but not necessarily conditionally identically distributed random variables is established. These are done anticipating that the field of conditional limit theory will prove to be of significant applicability.

Keywords

conditional independence;conditional identical distribution;conditional characteristic function;conditional central limit theorem;

Language

English

Cited by

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SOME RESULTS ON CONDITIONALLY UNIFORMLY STRONG MIXING SEQUENCES OF RANDOM VARIABLES,;;;

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References

1.

V. Basawa and B. L. S. Prakasa Rao, Statistical Inference for Stochastic Processes, London, Academic press, 1980.

2.

W. Grzenda and W.Zieba, Conditional central limit theorems, Int. Math. Forum 3 (2008), no. 29-32, 1521-1528.

3.

O. Kallenberg, Foundations of Modern Probability, 2nd Edition, Now York, Springer-Verlag, 2002.

4.

M. Loeve, Probability Theory II, 4th Edition, Now York, Springer-Verlag, 1978.

5.

D. Majerek, W. Nowak, and W. Zieba, Conditional strong law of large number, Int. J. Pure Appl. Math. 20 (2005), no. 2, 143-157.

6.

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.

7.

B. L. S. Prakasa Rao, Conditional independence, conditional mixing and conditional association, Ann. Inst. Statist. Math. 61 (2009), no. 2, 441-460.

8.

G. G. Roussas, On conditional independence, mixing, and association, Stoch. Anal. Appl. 26 (2008), no. 6, 1274-1309.

9.

A. N. Shiryaev, Probability, 2nd Edition, Now York, Springer-Verlag, 1995.

10.

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.

11.

D. M. Yuan and L. Lei, Some conditional results for conditionally strong mixing sequences of random variables, Sci. China Math. 56 (2013), no. 4, 845-859.