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

Yuping Zheng, Andrew Lamperski

TL;DR

The paper addresses non-asymptotic error analysis for classical non-parametric spectral estimators under $L$-mixing time-series data, deriving finite-sample bounds for the Bartlett and Welch methods. It develops a general stochastic-approximation framework and proves variance and bias bounds, plus high-probability results, that hold beyond Gaussian or ARMA-type assumptions and apply to a broad class of processes, including nonlinear and geometrically ergodic Markov chains. The key contributions are the non-asymptotic bounds with explicit dependence on mixing quantities, the extension of L-mixing properties to spectral data matrices, and simulations validating the theory on a finite-state Markov chain. The results bridge the gap between practical finite-sample spectral estimation and rigorous non-asymptotic guarantees, enabling more reliable performance assessment in dependent-data settings.

Abstract

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.

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

TL;DR

The paper addresses non-asymptotic error analysis for classical non-parametric spectral estimators under -mixing time-series data, deriving finite-sample bounds for the Bartlett and Welch methods. It develops a general stochastic-approximation framework and proves variance and bias bounds, plus high-probability results, that hold beyond Gaussian or ARMA-type assumptions and apply to a broad class of processes, including nonlinear and geometrically ergodic Markov chains. The key contributions are the non-asymptotic bounds with explicit dependence on mixing quantities, the extension of L-mixing properties to spectral data matrices, and simulations validating the theory on a finite-state Markov chain. The results bridge the gap between practical finite-sample spectral estimation and rigorous non-asymptotic guarantees, enabling more reliable performance assessment in dependent-data settings.

Abstract

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 -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 -mixing. In particular, -mixing processes can model a variety of nonlinear phenomena which do not satisfy the assumptions of our prior work. Our new error bounds for -mixing processes match the error bounds in the restrictive settings from prior work up to logarithmic factors.
Paper Structure (17 sections, 10 theorems, 29 equations, 2 figures)

This paper contains 17 sections, 10 theorems, 29 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. Problem Setup
  3. Notation
  4. $L$-mixing processes
  5. Problem and Algorithm
  6. Comparison with Assumptions from lamperski2023nonasymptotic
  7. Main Results of Convergence Analysis
  8. L-mixing Properties
  9. Variations on Classical $L$-mixing Results
  10. $L$-mixing properties of Spectral Data Matrices
  11. Proofs of Main Results
  12. Proof of Theorem \ref{['theorem_main']}
  13. Proof of Proposition \ref{['prop:bias']}
  14. Proof of Theorem \ref{['thm:highProb']}
  15. Proof of Proposition \ref{['prop:comparison']}
  16. ...and 2 more sections

Key Result

Proposition 1

If $\mathbf{y}[k]$ satisfies Aa:subgaussian and $h$ is causal with $\sum_{\ell=0}^{\infty}\|h[\ell]\|_2(\ell+1)<\infty$, then $\mathbf{y}[k]$ is $L$-mixing with

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 (15)

  • Proposition 1
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Remark 2
  • Proposition 2
  • Lemma 1
  • proof : Sketch
  • Lemma 2
  • ...and 5 more