On the central limit question under absolute regularity
Let \(X=(X_ k,k\in {\mathbb{Z}})\) be a strictly stationary sequence of random variables on a probability space (\(\Omega\),\({\mathfrak F},P)\). Denote \(S_ n=X_ 1+...+X_ n\), \(n=1,2,..\). \(\beta ({\mathfrak A},{\mathfrak B})=\sup \quad 2^{-1}\sum^{I}_{i=1}\sum^{J}_{j=1}| P(A_ i\cap B_ j)- P(A_ i)P(B_ j)|,\) where sup is taken over all pairs of partitions \(\{A_ 1,...,A_ I\}\) and \(\{B_ 1,...,B_ J\}\) of \(\Omega\) such that \(A_ i\in {\mathfrak A}\) \(\forall i\) and \(B_ j\in {\mathfrak B}\) \(\forall j\); \({\mathfrak A}\), \({\mathfrak B}\) being \(\sigma\)-fields. \(\beta (n)=\beta ({\mathfrak F}^ 0_{-\infty},{\mathfrak F}_ n^{\infty})\), \(n=1,2,..\). Let (1) \(0<Var X_ 0<\infty\), \(corr(X_ 0,X_ n)=0\) \(\forall n\geq 1\); (2) \(\inf_{n\geq 1}P(S_ n=0)>0\); (3) \(\lim_{c\to \infty}[\sup_{n\geq 1}P(| S_ n| >c)]=0.\) Theorem 3. Suppose \(\delta >0\). Then there exists a strictly stationary sequence \(X=(X_ k)\) such that \(EX_ 0=0\), \(E| X_ 0|^{2+\delta}<\infty\), \(\sum^{\infty}_{m=1}\beta (m)^{\delta /2+\delta}<\infty\), Var \(S_ m\sim (\log m)^{-4}\cdot m\) as \(m\to \infty\) and (2) and (3) both hold. Theorem 4. There exists a strictly stationary sequence \(X=(X_ k)\) such that \(EX_ 0=0\), \(| X_ 0| <C\) a.s. for some \(C<\infty\), \(\sum^{\infty}_{m=1}\beta (m)<\infty\), Var \(S_ m\sim (\log m)^{- 4}\cdot m\) as \(m\to \infty\), and (2) and (3) both hold.
scientific article; zbMATH DE number 3932118