Title: Non-Asymptotic Analysis of Classical Spectrum Estimators for $L$-mixing Time-series Data with Unknown Means Show Chinese title

Title: 非渐近分析经典谱估计器在未知均值的$L$-混合时间序列数据中的应用

Abstract: Spectral estimation is an important tool in time series analysis, with applications including economics, astronomy, and climatology. The asymptotic theory for non-parametric estimation is well-known but the development of non-asymptotic theory is still ongoing. Our recent work obtained the first non-asymptotic error bounds on the Bartlett and Welch methods for $L$-mixing stochastic processes. The class of $L$-mixing processes contains common models in time series analysis, including autoregressive processes and measurements of geometrically ergodic Markov chains. Our prior analysis assumes that the process has zero mean. While zero-mean assumptions are common, real-world time-series data often has unknown, non-zero mean. In this work, we derive non-asymptotic error bounds for both Bartlett and Welch estimators for $L$-mixing time-series data with unknown means. The obtained error bounds are of $O(\frac{1}{\sqrt{k}})$, where $k$ is the number of data segments used in the algorithm, which are tighter than our previous results under the zero-mean assumption. Show Chinese abstract

Abstract: 谱估计是时间序列分析中的一个重要工具，应用领域包括经济学、天文学和气候学。 非参数估计的渐近理论众所周知，但非渐近理论的发展仍在进行中。 我们最近的工作获得了针对 $L$-混合随机过程的巴特利特和韦尔奇方法的第一个非渐近误差界。 $L$-混合过程类包含了时间序列分析中常见的模型，包括自回归过程和几何遍历马尔可夫链的测量。 我们之前的研究假设该过程均值为零。 虽然均值为零的假设很常见，但真实世界的时间序列数据通常具有未知的非零均值。 在这项工作中，我们推导了针对均值未知的 $L$-混合时间序列数据的巴特利特和韦尔奇估计器的非渐近误差界。 所获得的误差界为 $O(\frac{1}{\sqrt{k}})$，其中 $k$ 是算法中使用的数据段数量，这些误差界比我们在零均值假设下的先前结果更紧密。