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 ${\mathcal{H}}_0$. In this framework, the usual univariate long memory parameter $d$ is replaced by a long memory operator $D$ acting on ${\mathcal{H}}_0$, leading to a class of ${\mathcal{H}}_0$-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 ${\mathcal{H}}_0$-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 ${\mathcal{H}}_0$ with a finite second moment, the empirical autocovariance operator converges, in trace-norm, to the true autocovariance operator almost surely at each lag.

分数积分自回归移动平均 (FIARMA) 过程已被广泛且成功地用于建模和预测表现出长程依赖性的单变量时间序列。最近还考虑了这些过程的矢量和功能扩展。在这里，我们在可分离希尔伯特空间中对过程进行评估的情况下，依靠谱域方法来研究这些过程${\mathcal{H}}_0$。在此框架中，通常的单变量长记忆参数$d$被长记忆运算符取代 $D$作用于${\mathcal{H}}_0$，导致一类${\mathcal{H}}_0$-FIARMA 估值（$D,p,q$) 过程，其中$p$和$q$是 AR 和 MA 多项式的次数。什么时候$D$是一个正常的经营者，我们提供了充分必要条件$D$- 分数阶积分${\mathcal{H}}_0$-ARMA 值（$p,q$）流程要明确定义。然后，我们推导一类因果 FIARMA 过程的最佳预测变量，并研究如何从过程的有限样本中一致地估计这个最佳预测变量。为此，我们提供了周期图的二次函数的一般结果，这顺便产生了独立兴趣的结果。也就是说，对于任何遍历平稳过程${\mathcal{H}}_0$在有限的二阶矩的情况下，经验自协方差算子在迹范数中几乎肯定在每个滞后处收敛到真正的自协方差算子。