ISSUES CONCERNING THE APPROXIMATION UNDERLYING THE SPECTRAL REPRESENTATION THEOREM
谱表示定理所用逼近的若干问题AI 解读 ↓
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AI 根据所提供的论文材料归类：摘要围绕平稳随机过程、谱表示和时间范围上的逼近性质展开，直接对应时间序列理论。
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谱表示定理所用逼近的若干问题
- 研究问题
- 将平稳随机过程用有限个随机加权谐波逼近时，误差是否平稳，且均方逼近能否在无界时间范围上一致成立？
- 方法与思路
- 作者分析谱表示的过程版本，考察有限谐波和的均方误差及其随时间变化的性质，区分有界与无界时间范围。
- 材料陈述的贡献
- 摘要称文中相关情形在无界时间范围上不具一致逼近，误差过程不平稳，因此过程版本应明确有界范围的限制。
- 阅读关注点
- 应核对“相关情形”的准确范围、均方与一致收敛的定义，以及有限谐波数量随时间和预设精度变化的论证。
- 阅读边界
- 仅据所给 RePEc 出版社摘要；具体过程条件与例外未说明，不能将结论无条件推广到所有平稳过程或其他版本。
原文摘要
In many important textbooks the formal statement of the spectral representation theorem is followed by a process version, usually informal, stating that any stationary stochastic process {ξ(t), t ∈ T} is the limit in quadratic mean of a sequence of processes {S(n,t), t ∈ T}, each consisting of a finite sum of harmonic oscillations with stochastic weights. The natural issues, whether the approximation error ξ(t) − S(n,t) is stationary or whether at least it converges to zero uniformly in t, have not been explicitly addressed in the literature. The paper shows that in all relevant cases, for T unbounded the process convergence is not uniform in t (so that ξ(t) − S(n,t) is not stationary). Equivalently, when T is unbounded the number of harmonic oscillations necessary to approximate ξ(t) with a preassigned accuracy depends on t. The conclusion is that the process version of the spectral representation theorem should explicitly mention that in general the approximation of ξ(t) by a finite sum of harmonic oscillations, given the accuracy, is valid for t belonging to a bounded subset of the real axis (of the set of integers in the discrete-parameter case).The author is grateful for very useful suggestions to Francesco Battaglia, Gianluca Cubadda, Domenico Marinucci, Enzo Orsingher, Dag Tjøstheim, and Umberto Triacca and also to an anonymous referee and the Econometric Theory co-editor Benedikt M. Pötscher.