Spectral estimation is a fundamental problem for time series analysis, which is widely applied in economics, speech analysis, seismology, and control systems. The asymptotic convergence theory for classical, non-parametric estimators, is well-understood, but the non-asymptotic theory is still rather limited. Our recent work gave the first non-asymptotic error bounds on the well-known Bartlett and Welch methods, but under restrictive assumptions. In this paper, we derive non-asymptotic error bounds for a class of non-parametric spectral estimators, which includes the classical Bartlett and Welch methods, under the assumption that the data is an $L$-mixing stochastic process. A broad range of processes arising in time-series analysis, such as autoregressive processes and measurements of geometrically ergodic Markov chains, can be shown to be $L$-mixing. In particular, $L$-mixing processes can model a variety of nonlinear phenomena which do not satisfy the assumptions of our prior work. Our new error bounds for $L$-mixing processes match the error bounds in the restrictive settings from prior work up to logarithmic factors.

翻译：谱估计是时间序列分析中的基本问题，广泛应用于经济学、语音分析、地震学与控制系统等领域。对于经典非参数估计器的渐近收敛理论已有充分研究，但其非渐近理论仍相当有限。我们近期工作首次给出了著名Bartlett与Welch方法的非渐近误差界，但需依赖较强假设条件。本文在数据为$L$混合随机过程的假设下，推导了一类非参数谱估计器（包含经典Bartlett与Welch方法）的非渐近误差界。时间序列分析中广泛存在的过程（如自回归过程与几何遍历马尔可夫链的观测值）均可证明具有$L$混合性。特别地，$L$混合过程能够模拟多种非线性现象，这些现象并不满足我们先前工作中的假设条件。针对$L$混合过程的新误差界，在对数因子范围内与先前工作在限制性设定下得到的误差界保持一致。