Theorems and fallacies in the theory of long-range-dependent Processes

It is frequently claimed in the literature that long-range dependence has equivalent formulations in the time domain and the frequency domain. Although many researchers understand that this is only "operationally true," i.e., it holds in cases of interest, many state this equivalence as a mathematical theorem. In particular, it is claimed as a theorem in the literature that if a covariance function decays like one over a fractional power of n, then the corresponding power spectral density tends to infinity at the origin. It is shown here that the power spectral density need not exist. Conversely, if the power spectral density exists and tends to infinity at the origin, it is shown here that the covariance may not have the claimed decay. To conclude, a new theorem is proved that gives sufficient conditions on the power spectral density to guarantee that a process is asymptotically second-order self-similar (ASOSS). This result is used to provide a counterexample to the claim in the literature that asymptotic second-order self-similarity implies long-range dependence