In this paper we establish the basic asymptotic theory for periodic moving averages of i.i.d. random variables with regularly varying tails. The moving average coefficients are allowed to vary according to the season. A simple reformulation yields the corresponding results for moving averages of random vectors. Our main result is that when the underlying random variables have finite variance but infinite fourth moment, the sample autocorrelations are asymptotically stable. It is well known in this case that sample autocorrelations in the classical stationary moving average model are asymptotically normal.
"Periodic moving averages of random variables with regularly varying tails." Ann. Statist. 25 (2) 771 - 785, April 1997. https://doi.org/10.1214/aos/1031833673