The central limit theorem for summability methods of some weakly dependent sequences (Q1088278)

Let \(\{\xi_ j\}_{j\in {\mathbb{Z}}}\) be a strictly stationary sequence of random variables with \(E\xi_ 0=0\); \(E\xi^ 2_ 0=1\); \({\mathbb{Z}}=\{0,\pm 1,\pm 2,...\}\). Let \(\{c_ j(\lambda)\), \(j=1,2,...\); \(\lambda\in [0,\infty)\}\) be real numbers such that \(c_ j(\lambda)\in [0,1]\) for all j and \(\lambda\), and \(\sum^{\infty}_{j=0}c_ j(\lambda)=1\) for all \(\lambda\). Define \(Z(\lambda)=\sum^{\infty}_{j=0}c_ j(\lambda)\xi_ j\), \(d^ 2(\lambda)=E| Z(\lambda)|^ 2\), \(Q(\lambda)=\sup_{j}c_ j(\lambda)/d(\lambda).\) First the authors prove that if \(\{\xi_ j\}\) is m-dependent (m\(\geq 1)\) and \(Q(\lambda)\to 0\) as \(\lambda\to \infty\) then the distribution of \(Z(\lambda)/d(\lambda)\) converges weakly to the standard normal distribution \(N(0,1)\) as \(\lambda\to \infty\) if either (i) \(E(\xi_ i\xi_ j)\geq 0\) for all i and j; or (ii) \(\sum^{m}_{j=2}| E(\xi_ 1\xi_ j)| <1/2\). This result extends a similar result of \textit{P. Embrechts} and \textit{M. Maejima} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 68, 191-204 (1984; Zbl 0535.60018)]. Further it is proved that a similar conclusion holds true for strongly \(\alpha\)-mixing sequences \(\{\xi_ j\}\) of random variables such that, for some \(\delta >0\), \(E| \xi_ 0|^{2+\delta}<\infty\), \(\sum^{\infty}_{1}\{\alpha (n)\}^{\delta /(2+\delta)}<\infty\), there exists a positive integer valued function \(m_{\lambda}=m(\lambda)\) such that \(m(\lambda)/Q(\lambda)\to 0\) and \(\alpha (m(\lambda))d^{-2}(\lambda)\to 0\) as \(\lambda\to \infty\) and which satisfy (i) or (ii).