On the asymptotic independence of the sum and rare values of weakly dependent stationary random variables

It is shown that if the stationary sequence {Xi} has finite variance and satisfies a certain mixing condition, then the asymptotic distribution of ∑ni=1 Xn is unaffected by the information of whether the summands are in certain “rare” sets. An application of the result shows that ∑ni=1 Xi and the extremes of X1,…,Xn are asymptotically independent. This is in sharp contrast to the infinite variance case.