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

Didier Dacunha-Castelle,Lisandro Fermin

doi:10.1214/ecp.v11-1133

Didier Dacunha-Castelle, Lisandro Fermin

Open Access

https://doi.org/10.1214/ecp.v11-1133

Copy DOI

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.

Highlights

Let X be a stochastic second-order stationary process with covariance function γ and density spectral F
We prove that a large set of long memory (LM) processes 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

Summary

Introduction

Let X be a stochastic second-order stationary process with covariance function γ and density spectral F. We consider that X is a given centered stationary process with spectral density F. We prove that a very large set of stationary processes, whose spectral densities have singularities of different kinds, can be disaggregated by involving processes whose spectral densities are in C∞ (or in CH , with H ∈ N). LM processes are included in this set, for instance processes whose spectral densities F have a single singularity where F and its derivatives are explicitly controlled by functions with exponential growth.

Disaggregation using Multiplicative Kernels

Continuous Densities Class

Analytic Spectral Densities Class

Case of Non Independent Elementary Processes