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Non-Asymptotic Analysis of Classical Spectrum Estimators for $L$-mixing Time-series Data with Unknown Means

Yuping Zheng, Andrew Lamperski

TL;DR

This work derives non-asymptotic error bounds for Bartlett and Welch spectral estimators on $L$-mixing time-series with unknown means, using both batch and online algorithms. The analysis shows $O\left(\frac{1}{\sqrt{k}}\right)$ convergence for the estimation error, with tighter constants than prior zero-mean results and accompanying high-probability bounds, by leveraging the $L$-mixing framework and transformations of the data. It also provides explicit bias bounds for Bartlett and Welch and demonstrates applicability to autoregressive and geometrically ergodic Markov-chain data, validated through simulations. The results advance non-asymptotic theory for non-parametric spectral estimation under practical conditions and suggest directions for tighter, frequency-dependent analyses and non-stationary extensions.

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.

Non-Asymptotic Analysis of Classical Spectrum Estimators for $L$-mixing Time-series Data with Unknown Means

TL;DR

This work derives non-asymptotic error bounds for Bartlett and Welch spectral estimators on -mixing time-series with unknown means, using both batch and online algorithms. The analysis shows convergence for the estimation error, with tighter constants than prior zero-mean results and accompanying high-probability bounds, by leveraging the -mixing framework and transformations of the data. It also provides explicit bias bounds for Bartlett and Welch and demonstrates applicability to autoregressive and geometrically ergodic Markov-chain data, validated through simulations. The results advance non-asymptotic theory for non-parametric spectral estimation under practical conditions and suggest directions for tighter, frequency-dependent analyses and non-stationary extensions.

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 -mixing stochastic processes. The class of -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 -mixing time-series data with unknown means. The obtained error bounds are of , where is the number of data segments used in the algorithm, which are tighter than our previous results under the zero-mean assumption.

Paper Structure

This paper contains 17 sections, 7 theorems, 27 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. Problem Setup
  3. Notation
  4. Problem and Algorithm
  5. Background on the class of $L$-mixing Processes
  6. $L$-mixing Properties of Transformed Data
  7. Convergence Analysis
  8. Convergence of Mean Estimate
  9. Main Results
  10. Proof of Supporting Lemmas
  11. Proof of Lemma \ref{['lem:L-mixing_constant_collection']}
  12. Proof of Lemma \ref{['lem:mean_estimate_error']}
  13. Proofs of main results
  14. Proof of Theorem \ref{['theorem_main_batch']}
  15. Proof of Theorem \ref{['theorem_main_online']}
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\mathbf{y}_k\in\mathcal{Y}$ be a zero-mean $L$-mixing discrete-time process and let $w_k\in\mathbb{C}$. For all $M\ge 1$ and all $q\ge 1$:

Figures (2)

  • Figure 1: Bartlett Estimator. $M =5, L = 10^7$
  • Figure 2: Welch Estimator. Hann Window, $M =16, K=8, L = 10^7$

Theorems & Definitions (9)

  • Theorem 1
  • Remark 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 2
  • Proposition 1