Gnedenko-Raikov’s Theorem Fails for Exchangeable Sequences

Affiliation(s)

Department of Mathematical Sciences, University of New Brunswick, Saint John, Canada.

Department of Mathematical Sciences, Lakehead University, Thunder Bay, Canada.

Department of Mathematical Sciences, University of New Brunswick, Saint John, Canada.

Department of Mathematical Sciences, Lakehead University, Thunder Bay, Canada.

ABSTRACT

We study the connection between the central limit theorem and law of large numbers for exchangeable sequences, and provide a counterexample to the Gnedenko-Raikov theorem for such sequences.

Cite this paper

G. Stoica and D. Li, "Gnedenko-Raikov’s Theorem Fails for Exchangeable Sequences,"*Applied Mathematics*, Vol. 3 No. 12, 2012, pp. 2019-2021. doi: 10.4236/am.2012.312A277.

G. Stoica and D. Li, "Gnedenko-Raikov’s Theorem Fails for Exchangeable Sequences,"

References

[1] A. Gut, “Gnedenko-Raikov’s Theorem, Central Limit Theory, and the Weak Law of Large Numbers,” Statistics and Probability Letters, Vol. 76, No. 17, 2006, pp. 1935-1939. doi:10.1016/j.spl.2006.04.042

[2] M. Klass and H. Teicher, “The Central Limit Theorem for Exchangeable Random Variables without Moments,” Annals of Probability, Vol. 15, No. 1, 1987, pp. 138-153. doi:10.1214/aop/1176992260

[3] G. Stoica and D. Li, “On the Kolmogorov-Feller Law for Exchangeable Random Variables,” Statistics and Probability Letters, Vol. 80, No. 9-10, 2012, pp. 899-902. doi:10.1016/j.spl.2010.01.025

[4] G. Stoica, “An Extension of the Weak Law of Large Numbers for Exchangeable Sequences,” Acta Applicandae Mathematicae, Vol. 109, No. 3, 2010, pp. 759-763. doi:10.1007/s10440-008-9344-x

[5] M. V. Kruglov, “A Generalization of the Weak Law of Large Numbers,” Stochastic Analysis and Applications, Vol. 29, No. 4, 2011, pp. 674-683. doi:10.1080/07362994.2011.581099

[6] X. Jiang and M. G. Hahn, “Empirical Central Limit Theorems for Exchangeable Random Variables,” Statistics and Probability Letters, Vol. 59, No. 1, 2002, pp. 75-81. doi:10.1016/S0167-7152(02)00204-3

[7] G. Stoica, “A Self-Normalized Weak Law of Large Numbers for Exchangeable Sequences,” Advances and Applications in Statistics, Vol. 25, No. 2, 2011, pp. 103-108.

[1] A. Gut, “Gnedenko-Raikov’s Theorem, Central Limit Theory, and the Weak Law of Large Numbers,” Statistics and Probability Letters, Vol. 76, No. 17, 2006, pp. 1935-1939. doi:10.1016/j.spl.2006.04.042

[2] M. Klass and H. Teicher, “The Central Limit Theorem for Exchangeable Random Variables without Moments,” Annals of Probability, Vol. 15, No. 1, 1987, pp. 138-153. doi:10.1214/aop/1176992260

[3] G. Stoica and D. Li, “On the Kolmogorov-Feller Law for Exchangeable Random Variables,” Statistics and Probability Letters, Vol. 80, No. 9-10, 2012, pp. 899-902. doi:10.1016/j.spl.2010.01.025

[4] G. Stoica, “An Extension of the Weak Law of Large Numbers for Exchangeable Sequences,” Acta Applicandae Mathematicae, Vol. 109, No. 3, 2010, pp. 759-763. doi:10.1007/s10440-008-9344-x

[5] M. V. Kruglov, “A Generalization of the Weak Law of Large Numbers,” Stochastic Analysis and Applications, Vol. 29, No. 4, 2011, pp. 674-683. doi:10.1080/07362994.2011.581099

[6] X. Jiang and M. G. Hahn, “Empirical Central Limit Theorems for Exchangeable Random Variables,” Statistics and Probability Letters, Vol. 59, No. 1, 2002, pp. 75-81. doi:10.1016/S0167-7152(02)00204-3

[7] G. Stoica, “A Self-Normalized Weak Law of Large Numbers for Exchangeable Sequences,” Advances and Applications in Statistics, Vol. 25, No. 2, 2011, pp. 103-108.