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http://dx.doi.org/10.4134/BKMS.2010.47.6.1139

SELF-NORMALIZED WEAK LIMIT THEOREMS FOR A ø-MIXING SEQUENCE  

Choi, Yong-Kab (DEPARTMENTS OF MATHEMATICS AND RINS GYEONGSANG NATIONAL UNIVERSITY)
Moon, Hee-Jin (DEPARTMENTS OF MATHEMATICS AND RINS GYEONGSANG NATIONAL UNIVERSITY)
Publication Information
Bulletin of the Korean Mathematical Society / v.47, no.6, 2010 , pp. 1139-1153 More about this Journal
Abstract
Let {$X_j,\;j\geq1$} be a strictly stationary $\phi$-mixing sequence of non-degenerate random variables with $EX_1$ = 0. In this paper, we establish a self-normalized weak invariance principle and a central limit theorem for the sequence {$X_j$} under the condition that L(x) := $EX_1^2I{|X_1|{\leq}x}$ is a slowly varying function at $\infty$, without any higher moment conditions.
Keywords
self-normalized random variables; invariance principle; central limit theorem; mixing sequence;
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1 A. Rackauskas and C. Suquet, Invariance principles for adaptive self-normalized partial sums processes, Stochastic Process. Appl. 95 (2001), no. 1, 63-81.   DOI   ScienceOn
2 Q. M. Shao, Almost sure invariance principles for mixing sequences of random variables, Stochastic Process. Appl. 48 (1993), no. 2, 319-334.   DOI   ScienceOn
3 Q. M. Shao, An invariance principle for stationary ${\rho}-mixing$ sequence with infinite variance, Chinese Ann. Math. Ser. B 14 (1993), no. 1, 27-42.
4 Q. M. Shao, Self-normalized large deviations, Ann. Probab. 25 (1997), no. 1, 285-328.   DOI   ScienceOn
5 W. Wang, Self-normalized lag increments of partial sums, Statist. Probab. Lett. 58 (2002), no. 1, 41-51.   DOI   ScienceOn
6 M. Peligrad, The convergence of moments in the central limit theorem for ${\rho}-mixing$ sequences of random variables, Proc. Amer. Math. Soc. 101 (1987), no. 1, 142-148.
7 M. Peligrad and Q. M. Shao, Estimation of the variance of partial sums for ${\rho}-mixing$ random variables, J. Multivariate Anal. 52 (1995), no. 1, 140-157.   DOI   ScienceOn
8 N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987.
9 R. Balan and R. Kulik, Weak invariance principle for mixing sequences in the domain of attraction of normal law, Studia Sci. Math. Hungarica 46 (2009), no. 3, 329-343.
10 R. Balan and I. M. Zamfirescu, Strong approximation for mixing sequences with infinite variance, Electron. Comm. Probab. 11 (2006), 11-23.   DOI
11 R. C. Bradley, A central limit theorem for stationary ${\rho}-mixing$ sequences with infinite variance, Ann. Probab. 16 (1988), no. 1, 313-332.   DOI   ScienceOn
12 M. Csorgo, Z. Y. Lin, and Q. M. Shao, Studentized increments of partial sums, Sci. China Ser. A. 37 (1994), no. 3, 265-276.
13 M. Csorgo, B. Szyszkowicz, and Q. Wang, Donsker's theorem for self-normalized partial sums processes, Ann. Probab. 31 (2003), no. 3, 1228-1240.   DOI   ScienceOn
14 N. Etemadi, On some classical results in probability theory, Sankhya Ser. A 47 (1985), no. 2, 215-221.
15 P. Griffin and J. Kuelbs, Some extensions of the LIL via self-normalizations, Ann. Probab. 19 (1991), no. 1, 380-395.   DOI   ScienceOn
16 Z. Y. Lin and C. R. Lu, Limit Theory for Mixing Dependent Random Variables, Kluwer Academic Publishers, Dordrecht; Science Press, New York, 1996.