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

Abstract

Let Xt = sum_{j=-\infty}^\infty c_j Z_{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 a and \alpha/2, respectively. This result immediately gives the limit distribution for the least squares estimates of the parameters in an autoregressive process.