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

Abstract

For j= 1, ..., J, let Kj : R -+ R be measurable bounded functions and Xj,,, = -fR aj(n cjx)M(dx), n > 1, be a-stable moving averages where a E (0, 2), cj > 0 for j= 1, ..., J, and M(dx) is an astable random measure on R with the Lebesgue control measure and skewness intensity f# G [-1, 1]. We provide conditions on the functions aj and Kj, j =1,..., J, for the normalized partial sums vector Nj 1/2 L Ni.-EjXjAi-L..iJ n=vector N1/2 (Kj(Xj, n) EK(Xn)), j = 1, ..., J, to be asymptotically normal as Nj 00. 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.