## Bernoulli

- Bernoulli
- Volume 9, Number 5 (2003), 833-855.

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

Vladas Pipiras and Murad S. Taqqu

#### 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.

#### Article information

**Source**

Bernoulli, Volume 9, Number 5 (2003), 833-855.

**Dates**

First available in Project Euclid: 17 October 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1066418880

**Digital Object Identifier**

doi:10.3150/bj/1066418880

**Mathematical Reviews number (MathSciNet)**

MR2047688

**Zentralblatt MATH identifier**

1053.60017

**Keywords**

central limit theorem moving averages stable distributions

#### Citation

Pipiras, Vladas; Taqqu, Murad S. Central limit theorems for partial sums of bounded functionals of infinite-variance\\moving averages. Bernoulli 9 (2003), no. 5, 833--855. doi:10.3150/bj/1066418880. https://projecteuclid.org/euclid.bj/1066418880