Central limit theorems for mixing sequences of random variables under minimal conditions (Q1083752)

Consider a strictly stationary, strongly mixing sequence of random variables \(\{X_ n,n\geq 1\}\) with \(EX_ 1=0\), \(EX^ 2_ 1=1\). Let \(S_ n=X_ 1+...+X_ n\), \(\rho_ n=(\pi /2)^{1/2}E| S_ n|\) and \(\sigma^ 2_ n=ES^ 2_ n\). It is shown that \(\{S_ n/\sigma_ n\}\) converges in distribution to the standard normal law if and only if \(\limsup_{n\to \infty} \sigma_ n/\rho_ n\leq 1\). Moreover, if \(\liminf_{n\to \infty} \rho_ n/\sigma_ n>0\) then \(\{S_{n_ k}/\rho_{n_ k}\}\) converges in distribution to the standard normal distribution for some sequence \(\{n_ k\}\). In both cases it is assumed that \(\sigma^ 2_ n=nL(n)\) for some slowly-varying function L.