## The Annals of Statistics

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

#### 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

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