Disaggregation of Long Memory Processes on C^(infty) 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^{infty}). We show that the C^{infty} class of spectral densities is the optimal class to get a general result for disaggregation of LM processes in SM processes, in the sense that the result given in C^{infty} class cannot be improved taking for instance analytic functions instead of indefinitely derivable functions.