## The Annals of Statistics

- Ann. Statist.
- Volume 14, Number 2 (1986), 533-558.

### Limit Theory for the Sample Covariance and Correlation Functions of Moving Averages

Richard Davis and Sidney Resnick

#### Abstract

Let $X_t = \sum^\infty_{j=-\infty} c_jZ_{t-j}$ be a moving average process where the $Z_t$'s are iid and have regularly varying tail probabilities with index $\alpha > 0$. The limit distribution of the sample covariance function is derived in the case that the process has a finite variance but an infinite fourth moment. Furthermore, in the infinite variance case $(0 < \alpha < 2)$, the sample correlation function is shown to converge in distribution to the ratio of two independent stable random variables with indices $\alpha$ and $\alpha/2$, respectively. This result immediately gives the limit distribution for the least squares estimates of the parameters in an autoregressive process.

#### Article information

**Source**

Ann. Statist., Volume 14, Number 2 (1986), 533-558.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176349937

**Digital Object Identifier**

doi:10.1214/aos/1176349937

**Mathematical Reviews number (MathSciNet)**

MR840513

**Zentralblatt MATH identifier**

0605.62092

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Secondary: 62E20: Asymptotic distribution theory 60F05: Central limit and other weak theorems

**Keywords**

Sample covariance and correlation functions regular variation stable laws moving average point processes

#### Citation

Davis, Richard; Resnick, Sidney. Limit Theory for the Sample Covariance and Correlation Functions of Moving Averages. Ann. Statist. 14 (1986), no. 2, 533--558. doi:10.1214/aos/1176349937. https://projecteuclid.org/euclid.aos/1176349937