A general central limit theorem for strong mixing sequences
Author's abtract: A central limit theorem for strong mixing sequences is given that applies to both nonstationary sequences and triangular array settings. The result improves on an earlier central limit theorem for this type of dependence given by \textit{D. N. Politis} et al. in [J. Econom. 81, No. 2, 281--317 (1997; Zbl 0904.62059)].NEWLINENEWLINENEWLINEBelow are some excerpts from the author's introduction:NEWLINENEWLINENEWLINEFor a survey of the literature on CLTs for mixing sequences we refer to, e.g., [\textit{P. Doukhan}, Mixing: Properties and examples. New York: Springer-Verlag (1994; Zbl 0801.60027); \textit{Z.-Y. Lin} and \textit{C.-R. Lu}, Limit theory for mixing dependent random variables. Dordrecht: Kluwer Academic Publishers. New York, NY: Science Press (1996; Zbl 0889.60001); \textit{R. C. Bradley}, Introduction to strong mixing conditions. Vol. 1. Heber City, UT: Kendrick Press (2007; Zbl 1133.60001); Vol. 2. Heber City, UT: Kendrick Press (2007; Zbl 1133.60002); Vol. 3. Heber City, UT: Kendrick Press (2007; Zbl 1134.60004)].NEWLINENEWLINENEWLINEIt should be noted that in addition to moment and mixing conditions, strict stationarity is assumed in many cases. An exception is Politis et al. [loc. cit.], who obtained a CLT which applies to both non-stationary sequences and triangular array settings. Let \(\{ X_{n,i}: 1\leq I \leq d_n \}\) be a triangular array of random variables and denote the strong mixing coefficient corresponding to the \(n\)th row by \(\alpha_{n}(\cdot)\). Define \(X_{n,d_{n}}=d^{-1}_{n}\sum^{d_{n}}_{i=1} X_{n,i}\), \({X}_{n,b,t} =b^{-1}\sum^{t+b-1}_{i=t} X_{n,i}\), and \(\sigma^{2}_{n,b,t}=\text{var}(b^{1/2}={X}_{n,b,t})\). Politis et al. [loc.\,cit., Theorem A.1; Subsampling. New York, NY: Springer (1999; Zbl 0931.62035), Theorem B.0.1] showed that \(d^{1/2}_{n}(=X_{n,d_{n}}-E ={X}_{n,d_{n}}) \) converges weakly to a normal distribution with mean zero and variance \(\sigma^{2}\) under the following conditions: for some \(\delta >0 \): NEWLINE{\parindent=9mmNEWLINE\begin{itemize}\item[(B1)]: \(\operatorname{E}| X_{n,i} - \operatorname{E}X_{n,i}|^{2+\delta} < c\), for some \(c > 0\) and all \(n, i, \)NEWLINE\item[(B2)]: for any sequence \(b_n\) that tends to infinity with \(n\), \( \sup_t | \sigma^2_{n, b_n, t} -\sigma^2 |\rightarrow 0\) as \(n\rightarrow\infty,\)NEWLINE\item[(B3)]: \(\sum^\infty_{k=0}(k+1)^2\alpha^{\frac{\delta}{4+\delta}}_n(k)<c\) for some \(c>0\) and all \(n\).NEWLINENEWLINE\end{itemize}}NEWLINEPolitis et al. [Zbl 0931.62035, p. 322] write that condition (B2) ``is not as restrictive as it might appear at first glance. In fact, it is difficult to imagine a reasonable situation where'' \(\sigma^{2}_{n, b_n, 1}\rightarrow \sigma^2\), ``but where condition (B2) is violated.'' In this paper, it is demonstrated that condition (B2) is superfluous, i.e., that a CLT is valid under conditions (B1) and (B3) only.
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