# Learning Ergodic Dynamical Systems from a Finite Trajectory

arXiv: https://arxiv.org/abs/2607.22399 · HTML: https://mlsanity.com/paper/2607.22399/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 142 references that are in the map.
- First posted on arXiv 24 Jul 2026.

## Builds on (most cited first, 7 of 7)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Theory and Algorithms for Forecasting Time Series](https://mlsanity.com/paper/1803.05814/) (2018)
- [An Introductory Guide to Koopman Learning](https://mlsanity.com/paper/2510.22002/) (2025)
- [On the Consistency of Kernel Methods with Dependent Observations](https://mlsanity.com/paper/2406.06101/) (2024)
- [Hoeffding's Inequality for Markov Chains under Generalized Concentrability Condition](https://mlsanity.com/paper/2310.02941/) (2023)
- [Nonparametric Sparse Online Learning of the Koopman Operator](https://mlsanity.com/paper/2405.07432/) (2024)
- [Online Learning for Nonlinear Dynamical Systems without the I.I.D. Condition](https://mlsanity.com/paper/2504.02995/) (2025)

## Built on by (most cited first, 1 of 1)

- [Markov Chain Convergence in Kernel-Induced Geometry](https://mlsanity.com/paper/2610.13916/) (2026)

## Abstract (arXiv)

We consider the problem of learning from a single finite trajectory of an ergodic stochastic dynamical system. More precisely, we study discrete-time autonomous stochastic systems defining time-homogeneous Markov processes. We first focus on estimating the optimal one-step prediction function by nonlinear least squares, and derive high-probability guarantees measured with respect to the invariant measure of the process. These results make explicit how the non-independent and non-identically distributed nature of trajectory data modifies the classical statistical learning analysis. We then extend the framework to higher-order systems and finite-state spaces. Finally, we show that the same least squares and concentration arguments naturally extend to learning Koopman operators. Our approach combines tools from statistical learning theory and quantitative ergodic theory for Markov chains. It relies, in particular, on a concentration inequality for Hilbert-space-valued additive functionals of uniformly geometrically ergodic Markov chains.