Central limit theorems for partial sums of bounded functionals of infinite-variance\ averages

For j=1, dots, J, let Kⱼ: R be measurable bounded functions and X₍, ₉ = Raⱼ (n-cⱼx) M (dx), n 1, be -stable moving averages where (0, 2), cⱼ>0 for j=1, dots, J, and M (dx) is an -stable random measure on R with the Lebesgue control measure and skewness intensity Berry--Esseen boundsta-1, 1. We provide conditions on the functions aⱼ and Kⱼ, j=1, dots, J, for the normalized partial sums vector Nⱼ^-1/2 ₍=₁^Nⱼ (Kⱼ (X₉, ₍) - EKⱼ (X₉, ₍) ), j=1, dots, J, to be asymptotically normal as Nⱼ. This extends a result established by Tailen Hsing in the context of causal moving averages with discrete-time stable innovations. We also consider the case of moving averages with innovations that are in the stable domain of attraction.