EXTENSIONS OF SEVERAL CLASSICAL RESULTS FOR INDEPENDENT AND IDENTICALLY DISTRIBUTED RANDOM VARIABLES TO CONDITIONAL CASES

- Journal title : Journal of the Korean Mathematical Society
- Volume 52, Issue 2, 2015, pp.431-445
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2015.52.2.431

Title & Authors

EXTENSIONS OF SEVERAL CLASSICAL RESULTS FOR INDEPENDENT AND IDENTICALLY DISTRIBUTED RANDOM VARIABLES TO CONDITIONAL CASES

Yuan, De-Mei; Li, Shun-Jing;

Yuan, De-Mei; Li, Shun-Jing;

Abstract

Extensions of the Kolmogorov convergence criterion and the Marcinkiewicz-Zygmund inequalities from independent random variables to conditional independent ones are derived. As their applications, a conditional version of the Marcinkiewicz-Zygmund strong law of large numbers and a result on convergence in for conditionally independent and conditionally identically distributed random variables are established, respectively.

Keywords

conditional independence;conditional identical distribution;conditional Kolmogorov convergence criterion;conditional Marcinkiewicz-Zygmund inequalities;conditional Marcinkiewicz-Zygmund strong law;

Language

English

Cited by

References

1.

T. K. Chandra, Laws of Large Numbers, New Delhi, Narosa Publishing House, 2012.

2.

Y. S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Mar- tingales, 3rd Edition, New York, Springer-Verlag, 1997.

3.

T. C. Christofides and M. Hadjikyriakou, Conditional demimartingales and related re- sults, J. Math. Analy. Appl. 398 (2013), no. 1, 380-391.

4.

A. Gut, Probability: A graduate course, 2nd Edition, New York, Springer-Verlag, 2013.

5.

J. C. Liu and B. L. S. Prakasa Rao, On conditional Borel-Cantelli lemmas for sequences of random variables, J. Math. Anal. Appl. 399 (2013), no. 1, 156-165.

6.

J. C. Liu and L. D. Zhang, Conditional Borel-Cantelli lemma and conditional strong law of large number, Acta Math. Appl. Sin. (Chinese Ser.) 37 (2014), no. 3, 537-546.

7.

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

8.

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.

9.

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

10.

R. Pyke and D. Root, On convergence in r-mean for normalized partial sums, Ann. Math. Statist. 39 (1968), no. 2, 379-381.

11.

H. P. Rosenthal, On the subspaces of $L^p$ (p > 2) spanned by sequences of independent random variables, Israel J. Math. 8 (1970), no. 3, 273-303.

12.

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

13.

A. N. Shiryaev, Probability, 2nd Edition, New York, Springer-Verlag, 1996.

14.

X. H. Wang and X. J. Wang, Some inequalities for conditional demimartingales and conditional N-demimartingales, Statist. Probab. Lett. 83 (2013), no. 3, 700-709.

15.

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.

16.

D. M. Yuan, X. M. Hu, and B. Tao, Some results on conditionally uniformly strong mixing sequences of random variables, J. Korean Math. Soc. 51 (2014), no. 3, 609-633.

17.

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.

18.

D. M. Yuan, L. R. Wei, and L. Lei, Conditional central limit theorems for a sequence of conditional independent random variables, J. Korean Math. Soc. 51 (2014), no. 1, 1-15.