Sample correlations of infinite variance time series models: an empirical and theoretical study

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.