The sample autocorrelations of heavy-tailed processes with applications to ARCH

We study the sample ACVF and ACF of a general stationary sequence under a weak mixing condition and in the case that the marginal distributions are regularly varying. This includes linear and bilinear processes with regularly varying noise and ARCH processes, their squares and absolute values. We show that the distributional limits of the sample ACF can be random, provided that the variance of the marginal distribution is infinite and the process is nonlinear. This is in contrast to infinite variance linear processes. If the process has a finite second but infinite fourth moment, then the sample ACF is consistent with scaling rates that grow at a slower rate than the standard $\sqrt{n}$. Consequently, asymptotic confidence bands are wider than those constructed in the classical theory. We demonstrate the theory in full detail for an ARCH (1) process.