Hilbert valued fractionally integrated autoregressive moving average processes with long memory operators

Abstract

Fractionally integrated autoregressive moving average processes have been widely and successfully used to model univariate time series exhibiting long range dependence. Vector and functional extensions of these processes have also been considered more recently. Here we rely on a spectral domain approach to extend this class of models in the form of a general Hilbert valued processes. In this framework, the usual univariate long memory parameter d is replaced by a long memory operator D acting on the Hilbert space. Our approach is compared to processes defined in the time domain that were previously introduced for modeling long range dependence in the context of functional time series.