Journal of Applied Mathematics and Stochastic Analysis
Volume 11 (1998), Issue 3, Pages 255-282
doi:10.1155/S1048953398000227
Sample correlations of infinite variance time series models: an empirical and theoretical study
Cornell University, School of Operations Research and Industrial Engineering, Ithaca 14853, New York, USA
Received 1 September 1997; Revised 1 January 1998
Copyright © 1998 Jason Cohen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
When the elements of a stationary ergodic time series have finite variance
the sample correlation function converges (with probability 1) to the theoretical correlation function. What happens in the case where the variance
is infinite? In certain cases, the sample correlation function converges in
probability to a constant, but not always. If within a class of heavy tailed
time series the sample correlation functions do not converge to a constant,
then more care must be taken in making inferences and in model selection
on the basis of sample autocorrelations. We experimented with simulating
various heavy tailed stationary sequences in an attempt to understand
what causes the sample correlation function to converge or not to converge
to a constant. In two new cases, namely the sum of two independent moving averages and a random permutation scheme, we are able to provide
theoretical explanations for a random limit of the sample autocorrelation
function as the sample grows.