论文标题
希尔伯特（Hilbert
Hilbert valued fractionally integrated autoregressive moving average processes with long memory operators
论文作者
论文摘要
部分集成的自回旋运动平均值（FIARMA）过程已被广泛，成功地用于建模和预测表现出长距离依赖性的单变量时间序列。最近也考虑了这些过程的向量和功能扩展。在这里，我们通过依靠光谱域方法来研究这些过程，如果在可分离的希尔伯特空间H0中重视该过程的情况下。在此框架中，通常的单变量内存参数d被作用于H0的长存储器d代替，导致一类H0值fiarma（D，P，Q）过程，其中P和Q是AR和MA多项式的程度。当D是普通运算符时，我们为H0值ARMA（P，Q）过程的D分数整合提供了必要的条件。然后，我们得出了一类因果关系过程的最佳预测指标，并研究如何从该过程的有限样本中始终如一地估算该最佳预测指标。为此，我们为期刊的二次功能提供了一般的结果，该功能偶然产生了独立利益的结果。也就是说，对于有限第二刻（有限的第二刻）中价值有限的任何千古式固定过程，经验自动助行操作员以痕量为单位将几乎每个滞后的真正自动variance操作员融合到了真正的自动variance操作员。
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.
