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.
Publié le : 1986-06-14
Classification:
Sample covariance and correlation functions,
regular variation,
stable laws,
moving average,
point processes,
62M10,
62E20,
60F05
@article{1176349937,
author = {Davis, Richard and Resnick, Sidney},
title = {Limit Theory for the Sample Covariance and Correlation Functions of Moving Averages},
journal = {Ann. Statist.},
volume = {14},
number = {2},
year = {1986},
pages = { 533-558},
language = {en},
url = {http://dml.mathdoc.fr/item/1176349937}
}
Davis, Richard; Resnick, Sidney. Limit Theory for the Sample Covariance and Correlation Functions of Moving Averages. Ann. Statist., Tome 14 (1986) no. 2, pp. 533-558. http://gdmltest.u-ga.fr/item/1176349937/