Chikin, D. O. Martingale methods for proving central limit theorems for dependent variables. (English) Zbl 0741.60015 Statistics and control of stochastic processes. Vol. 2, Pap. Steklov Semin., Moscow/USSR 1985-86, Transl. Ser. Math. Eng., 41-67 (1989). [For the entire collection see Zbl 0732.00014.]Let \(\{X_ n,\;-\infty<n<+\infty\}\) be a strictly stationary sequence of random variables with \(E X_ 0=0\), \(E X^ 2_ 0<\infty\). Define \({\mathcal F}_ n\) to be the sigma-field generated by \(\{X_ i:\;i\leq n\}\), and let \({\mathcal T}\) be a shift- invariant sigma-field such that \({\mathcal T}\subset{\mathcal F}_ 0\). Conditions are given which ensure that, for \(\alpha>1/2\), \[ E(\exp(itn^{-\alpha-1/2}\sum_{k=1}^ n k^ \alpha X_ k) \mid {\mathcal T})\to \exp\{-t^ 2 \sigma^ 2/2\}, \] where \(\sigma^ 2=\sum^ \infty_{n=-\infty}E(X_ n X_ 0 \mid {\mathcal T})\). In particular, this convergence is shown to hold for certain stationary mixing sequences. These results are derived from a more general theorem which establishes a similar convergence for weighted sums \(\sum^ n_{k=1}a_ k X_ k\), where \(a_ k=k^ \alpha f(k)\) and \(\{f(k)\}\) is slowly oscillating. Reviewer: R.J.Tomkins (Regina) MSC: 60F05 Central limit and other weak theorems 60G42 Martingales with discrete parameter 60G10 Stationary stochastic processes Keywords:strictly stationary sequence; invariant sigma-field; stationary mixing sequences Citations:Zbl 0732.00014 PDFBibTeX XMLCite \textit{D. O. Chikin}, in: Control of a diffusion process in a region with fixed reflection on the boundary. . 41--67 (1989; Zbl 0741.60015)