Spectral representations of weakly stationary processes valued in a separable Hilbert space : a survey with applications on functional time series
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In this paper, we review and clarify the construction of a spectral theory for weakly-stationary processes valued in a separable Hilbert space. We emphasize the link with functional analysis and provide thorough discussions on the different approaches leading to fundamental results on representations in the spectral domain. The clearest and most complete way to view such representations relies on a Gramian isometry between the time domain and the spectral domain. This theory is particularly useful for modeling functional time series. In this context, we define time invariant operator-valued linear filters in the spectral domain and derive results on composition and inversion of such filters. The advantage of a spectral domain approach over a time domain approach is illustrated through the construction of a class of functional autoregressive fractionaly integrated moving average processes which extend the celebrated class of ARFIMA processes that have been widely and successfully used to model univariate time series. Such functional ARFIMA processes are natural counterparts to processes defined in the time domain that were previously introduced for modeling long range dependence in the context of functional time series.
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