Weak convergence of partial sums of absolutely regular sequences (Q1058228)

Suppose \((X_ 1, X_ 2,...)\) is a stationary process. Define \(\beta(n)\) as \(E\{\sup_{A in B^{k+n+1}}^\infty| P(A| B^ k_ 1)-P(A)| \}\), where \(B^ m_ n\) denotes the \(\sigma\)-algebra generated by the variables \((X_ n,...,X_ m)\). If \(\lim_{n\to \infty}\beta (n)=0\), the process is called absolutely regular. Suppose \((X_ 1, X_ 2,...)\) is a stationary absolutely regular sequence of real-valued random variables with zero mean and \((2+\delta)\)-th absolute moment, where \(\delta >0\). Suppose \(\beta (n)<<n^{-(1+\epsilon)(1+2/\delta)}\) for some \(\epsilon >0\). Let \(S_ n\) denote \(X_ 1+...+X_ n\). Then \(\lim_{n\to \infty}n^{-1}Var(S_ n)\) exists: denote this limit by \(\sigma^ 2\). If \(\sigma^ 2>0\), it is shown that the asymptotic distribution of (n \(\sigma\) \({}^ 2)^{- 1/2}S_ n\) is the standard normal distribution.