On the asymptotic independence of the sum and rare values of weakly dependent stationary random variables (Q1909952)

The author investigates the joint behavior of the sum and extreme order statistics of random variables in a large sample from many different angles. In the infinite variance case (in particular in the heavy tailed case) the extreme order statistics of large sample \(X_1, \dots, X_n\) dominate \(S_n\), therefore it is possible to describe the asymptotic distribution \(S_n\) in terms of those of the extreme order statistics, however it is possible for the lower and upper extremes to cancel one another in the limit so that the sum is asymptotically independent of the extremes. The author shows that if the stationary sequence \(\{X_i\}\) has finite variance and satisfies strong mixing condition, then the asymptotic distribution of \(S_n\) is unaffected by the information of whether the summands are in certain ``rare'' sets. An application of the result shows that \(S_n\) and the extremes are asymptotically independent. This is in sharp contrast to the infinite variance case.