# Non-Asymptotic Pointwise and Worst-Case Bounds for Classical Spectrum Estimators

arXiv: https://arxiv.org/abs/2303.11908 · HTML: https://mlsanity.com/paper/2303.11908/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 27 references that are in the map.
- First posted on arXiv 21 Mar 2023.

## Builds on (most cited first, 10 of 10)

- [Near optimal finite time identification of arbitrary linear dynamical systems](https://mlsanity.com/paper/1812.01251/) (2018)
- [Gradient Descent Learns Linear Dynamical Systems](https://mlsanity.com/paper/1609.05191/) (2016)
- [Non-asymptotic Identification of LTI Systems from a Single Trajectory](https://mlsanity.com/paper/1806.05722/) (2018)
- [Finite Sample Analysis of Stochastic System Identification](https://mlsanity.com/paper/1903.09122/) (2019)
- [Non-asymptotic Closed-Loop System Identification using Autoregressive Processes and Hankel Model Reduction](https://mlsanity.com/paper/1909.02192/) (2019)
- [Spectral Analysis of High-dimensional Time Series](https://mlsanity.com/paper/1810.11223/) (2018)
- [Frequency Domain Statistical Inference for High-Dimensional Time Series](https://mlsanity.com/paper/2206.02250/) (2022)
- [Topology Learning of Linear Dynamical Systems with Latent Nodes using Matrix Decomposition](https://mlsanity.com/paper/1912.07152/) (2019)
- [Efficient and passive learning of networked dynamical systems driven by non-white exogenous inputs](https://mlsanity.com/paper/2110.00852/) (2021)
- [Statistical inference for high-dimensional spectral density matrix](https://mlsanity.com/paper/2212.13686/) (2022)

## Built on by (most cited first, 4 of 4)

- [Multi-Source and Test-Time Domain Adaptation on Multivariate Signals using Spatio-Temporal Monge Alignment](https://mlsanity.com/paper/2407.14303/) (2024)
- [Approximation with Random Shallow ReLU Networks with Applications to Model Reference Adaptive Control](https://mlsanity.com/paper/2403.17142/) (2024)
- [Information Theoretically Optimal Sample Complexity of Learning Dynamical Directed Acyclic Graphs](https://mlsanity.com/paper/2308.16859/) (2023)
- [Simultaneous Pointwise Majorization for Mixed Tail Processes with Applications in Gaussian Chaos and Ergodic Diffusions](https://mlsanity.com/paper/2609.01576/) (2026)

## Abstract (arXiv)

Spectrum estimation is a fundamental methodology in the analysis of time-series data, with applications including medicine, speech analysis, and control design. The asymptotic theory of spectrum estimation is well-understood, but the theory is limited when the number of samples is fixed and finite. This paper gives non-asymptotic error bounds for a broad class of spectral estimators, both pointwise (at specific frequencies) and in the worst case over all frequencies. The general method is used to derive error bounds for the classical Blackman-Tukey, Bartlett, and Welch estimators. In particular, these are first non-asymptotic error bounds for Bartlett and Welch estimators.