  This study aims to develop the limit theorems on the sample autocovariances
and sample autocorrelations for certain stationary infinitely divisible
processes. We consider the case where the infinitely divisible process has
heavy tail marginals and is generated by a conservative flow. Interestingly,
the growth rate of the sample autocovariances is determined by not only heavy
tailedness of the marginals but also memory length of the process. Although
this feature was first observed by \cite{resnick:samorodnitsky:xue:2000} for
some very specific processes, we will propose a more general framework from the
viewpoint of infinite ergodic theory. Consequently, the asymptotics of the
sample autocovariances can be more comprehensively discussed.
