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This study aim to develop 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 not only by heavy tailedness of the marginals but also by the memory length of the process. Although this feature was first observed by Resnick et al. (Stoch Process Appl 85:321–339, 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.

Limit Theory for the Sample Autocovariance for Heavy-Tailed Stationary Infinitely Divisible Processes Generated by Conservative Flows