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

Abstract

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 $H_0$. In this framework, the usual univariate long memory parameter $d$ is replaced by a long memory operator $D$ acting on $H_0$, leading to a class of $H_0$-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 $H_0$-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 $H_0$ with a finite second moment, the empirical autocovariance operator converges, in trace-norm, to the true autocovariance operator almost surely at each lag.