Disaggregation of Long Memory Processes on $\mathcal{C}^{\infty}$ Class

Abstract

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 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 $C^\infty$ class cannot be improved by taking for instance analytic functions instead of indefinitely derivable functions.