Central limit theorem for weakly dependent variables
Let \(X_ n\), \(n=0,\pm 1,..\). be a stationary random sequence with E \(X_ 1=0\), Var \(X_ 1<\infty\), \(S_ n=X_ 1+...+X_ n\), \(S_ n^*=S_ n/\sqrt{Var S_ n}\), \({\mathcal M}_ k=\sigma \{X_ i\), \(| i| \geq k\}\), satisfying the following conditions: 1) for all integers i, \(\rho_ i(n)=E| E(X_ 0X_ i| {\mathcal M}_ n)-E X_ 0X_ i| \to 0\), \(n\to \infty,\) 2) \(\sum^{\infty}_{k=-\infty}E X_ 0X_ k<\infty\) uniformly over all \(\bar X{}_ i\), \(i=0,\pm 1,....\) If \(\sigma^ 2=\sum^{\infty}_{k=-\infty}E X_ 0X_ k\neq 0\), then \[ \lim_{n\to \infty}P\{S^ x_ n<x\}\to (2\pi)^{-}\int^{x}_{- \infty}e^{-u^ 2/2}du. \]
scientific article; zbMATH DE number 4052676