New paper proves learning ergodic systems from a single trajectory

Researchers Oleksii Kachaiev, Silvia Villa, and Lorenzo Rosasco have tackled a fundamental challenge in dynamical systems: learning from a single finite trajectory of an ergodic stochastic process. Their paper, published on arXiv, focuses on discrete-time autonomous stochastic systems that form time-homogeneous Markov processes. They derive high-probability guarantees for the optimal one-step prediction function estimated via nonlinear least squares, measured under the process's invariant measure. This explicitly accounts for the non-independent and non-identically distributed nature of trajectory data, modifying classical statistical learning analysis.

The framework is extended to higher-order systems and finite-state spaces, and crucially, the same least-squares and concentration arguments naturally apply to learning Koopman operators—linear representations of nonlinear dynamics. The methodology relies on a concentration inequality for Hilbert-space-valued additive functionals of uniformly geometrically ergodic Markov chains. This work bridges statistical learning theory and quantitative ergodic theory, offering rigorous foundations for predictive modeling from limited sequential data, with potential applications in robotics, financial modeling, and climate analysis.