关于函数空间上时间序列的Herglotz定理的一个注记
A note on Herglotz's theorem for time series on function spaces
摘要：
本文证明了Hilbert值时间序列的Herglotz定理。这需要算子值测度的概念，我们将使它在我们的设定中精确。函数时间序列的Herglotz定理可以推广现有的结果，这些结果是函数空间上的频域分析的核心。特别地，我们利用这个结果证明了一大类过程的函数Cram{é}r表示的存在性，包括那些在谱分布和长记忆过程中具有跳跃的过程.此外，我们得到了一个最佳的有限维减少的时间序列在较弱的假设下比现有的文献。因此，本文的结果使傅里叶分析的过程，谱密度算子不一定存在。
英文摘要：
In this article, we prove Herglotz's theorem for Hilbert-valued time series. This requires the notion of an operator-valued measure, which we shall make precise for our setting. Herglotz's theorem for functional time series allows to generalize existing results that are central to frequency domain analysis on the function space. In particular, we use this result to prove the existence of a functional Cram{é}r representation of a large class of processes, including those with jumps in the spectral distribution and long-memory processes. We furthermore obtain an optimal finite dimensional reduction of the time series under weaker assumptions than available in the literature. The results of this paper therefore enable Fourier analysis for processes of which the spectral density operator does not necessarily exist.
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