Disaggregation of Long Memory Processes on C∞ Class

We prove that a large set of long memory (LM) processes (including classical LM processes and all processes whose spectral densities have a countable number of singularities controlled by exponential functions) are obtained by an aggregation procedure involving short memory (SM) processes whose spectral densities are infinitely differentiable (C∞). We show that the C∞ class of spectral densities infinitely differentiable is the best class to get a general result for disaggregation of LM processes in SM processes, in the sense that the result given in C∞ class cannot be improved by taking for instance analytic functions instead of indefinitely derivable functions.