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

Title & Authors
SELF-NORMALIZED WEAK LIMIT THEOREMS FOR A ø-MIXING SEQUENCE
Choi, Yong-Kab; Moon, Hee-Jin;

Abstract
Let {$\small{X_j,\;j\geq1}$} be a strictly stationary $\small{\phi}$-mixing sequence of non-degenerate random variables with $\small{EX_1}$ = 0. In this paper, we establish a self-normalized weak invariance principle and a central limit theorem for the sequence {$\small{X_j}$} under the condition that L(x) := $\small{EX_1^2I{|X_1|{\leq}x}}$ is a slowly varying function at $\small{\infty}$, without any higher moment conditions.
Keywords
self-normalized random variables;invariance principle;central limit theorem;mixing sequence;
Language
English
Cited by
1.
A self-normalized central limit theorem for a ρ-mixing stationary sequence, Communications in Statistics - Theory and Methods, 2017, 0
2.
A self-normalized invariance principle for a ϕ-mixing sequence, Periodica Mathematica Hungarica, 2013, 66, 2, 149
References
1.
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.

2.
R. Balan and I. M. Zamfirescu, Strong approximation for mixing sequences with infinite variance, Electron. Comm. Probab. 11 (2006), 11-23.

3.
N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987.

4.
R. C. Bradley, A central limit theorem for stationary ${\rho}-mixing$ sequences with infinite variance, Ann. Probab. 16 (1988), no. 1, 313-332.

5.
M. Csorgo, Z. Y. Lin, and Q. M. Shao, Studentized increments of partial sums, Sci. China Ser. A. 37 (1994), no. 3, 265-276.

6.
M. Csorgo, B. Szyszkowicz, and Q. Wang, Donsker's theorem for self-normalized partial sums processes, Ann. Probab. 31 (2003), no. 3, 1228-1240.

7.
N. Etemadi, On some classical results in probability theory, Sankhya Ser. A 47 (1985), no. 2, 215-221.

8.
P. Griffin and J. Kuelbs, Some extensions of the LIL via self-normalizations, Ann. Probab. 19 (1991), no. 1, 380-395.

9.
Z. Y. Lin and C. R. Lu, Limit Theory for Mixing Dependent Random Variables, Kluwer Academic Publishers, Dordrecht; Science Press, New York, 1996.

10.
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.

11.
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.

12.
A. Rackauskas and C. Suquet, Invariance principles for adaptive self-normalized partial sums processes, Stochastic Process. Appl. 95 (2001), no. 1, 63-81.

13.
Q. M. Shao, Almost sure invariance principles for mixing sequences of random variables, Stochastic Process. Appl. 48 (1993), no. 2, 319-334.

14.
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.

15.
Q. M. Shao, Self-normalized large deviations, Ann. Probab. 25 (1997), no. 1, 285-328.

16.
W. Wang, Self-normalized lag increments of partial sums, Statist. Probab. Lett. 58 (2002), no. 1, 41-51.