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

Abstract

For $j=1,\rm dots,J$, let $K_j:\mathbb{R}\to\mathbb{R}$ be measurable bounded functions and $X_{n,j} = \int_\mathbb{R}a_j(n-c_jx)M(\rm dx)$, $n\ge 1$, be $\alphapha$-stable moving averages where $\alphapha\in(0,2)$, $c_j>0$ for $j=1,\rm dots,J$, and $M(\rm dx)$ is an $\alphapha$-stable random measure on $\mathbb{R}$ with the Lebesgue control measure and skewness intensity $Berry--Esseen boundsta\in[-1,1]$. We provide conditions on the functions $a_j$ and $K_j$, $j=1,\rm dots,J$, for the normalized partial sums vector $ N_j^{-1/2} \sum_{n=1}^{N_j} (K_j(X_{j,n})-\rm EK_j (X_{j,n}))$, $j=1,\rm dots,J$, to be asymptotically normal as $N_j\to\infty$. 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.