  Fractionally integrated autoregressive moving average (FIARMA) processes have
been widely and successfully used to model and predict univariate time series
exhibiting long range dependence. Vector and functional extensions of these
processes have also been considered more recently. Here we study these
processes by relying on a spectral domain approach in the case where the
processes are valued in a separable Hilbert space H0. In this framework, the
usual univariate long memory parameter d is replaced by a long memory operator
D acting on H0, leading to a class of H0-valued FIARMA(D, p, q) processes,
where p and q are the degrees of the AR and MA polynomials. When D is a normal
operator, we provide a necessary and sufficient condition for the D-fractional
integration of an H0-valued ARMA(p, q) process to be well defined. Then, we
derive the best predictor for a class of causal FIARMA processes and study how
this best predictor can be consistently estimated from a finite sample of the
process. To this end, we provide a general result on quadratic functionals of
the periodogram, which incidentally yields a result of independent interest.
Namely, for any ergodic stationary process valued in H0 with finite second
moment, the empirical autocovariance operator converges, in trace-norm, to the
true autocovariance operator almost surely at each lag.
