prove ergodicity

This work addresses the lack of statistical fidelity in time-series modeling of dynamical systems, where conventional generalization error fails to guarantee accurate reproduction of physical invariants—such as invariant measures and Lyapunov exponents. To this end, we establish, for the first time, an ergodic-theoretic statistical generalization framework, explicitly identifying preservation of the physical measure as the central learning objective. We uncover a fundamental cause of statistical distortion in Neural ODEs applied to dynamical system regression: their neglect of Jacobian structural constraints. We theoretically prove—and empirically validate—that Jacobian regularization substantially enhances statistical generalization. On benchmark chaotic systems (Lorenz, Rössler), incorporating Jacobian information reduces estimation errors in invariant measures, statistical moments, and Lyapunov exponents by an average of 47%, outperforming MLPs, ResNets, Fourier Neural Networks, and RNNs.

This study addresses ambiguity in both the initial distribution and transition mechanism of continuous-time Markov processes. To jointly model these two sources of uncertainty, we introduce the novel concept of *imprecise Markov semigroups*. We establish geometric and topological ergodicity criteria—applicable to Euclidean spaces, Riemannian manifolds, and general measurable spaces—and rigorously prove sufficient conditions for exponential decay of uncertainty over time. Our methodology integrates convex analysis, operator semigroup theory, differential geometry, and measure theory. This work constitutes the first extension of classical ergodicity theory to settings with imprecise probabilistic specifications. The resulting framework provides a verifiable theoretical foundation and a unified analytical toolset for robust machine learning and uncertainty-aware visual modeling.

This paper addresses the estimation of β-mixing coefficients for geometrically ergodic Markov processes, aiming to quantify temporal dependence from a single observed trajectory. We propose a nonparametric estimation framework grounded in kernel density estimation and statistical learning theory, integrating tools from Besov space analysis, mixing process theory, and concentration inequalities. Our main contributions are threefold: (1) Under Besov-type regularity assumptions on the stationary density, we establish the first convergence rate with an explicit logarithmic correction factor; (2) For finite-state-space Markov chains, we achieve the optimal expected error rate $O(log n / sqrt{n})$ and corresponding high-probability bounds—without requiring any density boundedness or smoothness assumptions; (3) In continuous state spaces, we derive an expected error rate of $O(log n cdot n^{-s/(2s+2)})$, where $s > 0$ denotes the Besov smoothness index, along with matching high-probability guarantees.

Classical probabilistic constructions—such as almost-invariant σ-algebras, ergodic decompositions, the de Finetti theorem, and the zero–one law—lack a unified structural explanation within standard probability theory. Method: We construct a category whose objects are probability spaces and whose morphisms are measure-preserving Markov kernels, identified up to almost-sure equality. Our approach integrates categorical methods (notably the Markov category and its dagger structure), standard Borel space theory, and null-isomorphism techniques. Contribution/Results: We provide the first structural, categorical proof of the ergodic decomposition theorem; characterize almost-invariant σ-algebras uniformly as both limits and colimits; establish a dual limit–colimit characterization of invariant structures in random dynamical systems; and unify three foundational limit theorems—the de Finetti theorem, the zero–one law, and the ergodic decomposition—within a single abstract categorical framework. This advances the structural coherence, universality, and intrinsic categorical nature of probability theory.

This paper investigates the convergence of constant-step-size stochastic approximation (SA) algorithms under Markovian noise, focusing on root-finding—i.e., solving (f( heta^*) = 0)—and precisely characterizing the inherent bias and covariance error. To overcome the limitations of the classical i.i.d. noise assumption, we propose a joint parameter-perturbation process framework grounded in geometric ergodicity. This enables the first systematic analysis revealing a non-zero steady-state bias induced by “memory effects” in Markov noise, for which we derive a closed-form expression. Concurrently, we establish an explicit (O(alpha)) upper bound on the covariance error, quantifying how Markov dependence amplifies estimation error. Our theoretical results rigorously apply to temporal modeling settings such as TD-learning, and are corroborated by numerical experiments.

This work addresses the statistical learning challenges posed by non-i.i.d. data arising from a single finite trajectory of an ergodic stochastic dynamical system, focusing on one-step-ahead prediction modeling. By employing nonlinear least squares to estimate the predictive function and leveraging the invariant measure and uniform geometric ergodicity of the underlying Markov process, the study establishes the first high-probability generalization error bound for non-i.i.d. trajectories grounded in the system’s invariant measure. The approach integrates concentration inequalities for Hilbert space-valued additive functionals with Koopman operator approximation. The theoretical guarantees apply broadly across settings including higher-order systems and finite state spaces, thereby significantly extending the applicability of statistical learning theory to dynamical systems.

This work addresses the challenges in verifying the global minorization condition and obtaining initial-state-independent geometric convergence guarantees in Markov chain convergence analysis. To overcome these difficulties, the authors propose a novel framework that integrates a uniform drift condition with a local minorization condition, thereby establishing a stronger notion of hyper-V uniform ergodicity. This ensures that the deviations of all functions dominated by a Lyapunov function V converge to the invariant measure at a geometric rate. The approach circumvents the need to verify traditional global minorization conditions and instead leverages the structure of the invariant measure to directly infer qualitative hyper-V uniform ergodicity for two-variable Gibbs samplers. Moreover, it yields minimax-optimal convergence bounds. Empirical studies on Pólya–Gamma and Kolmogorov–Gamma Gibbs samplers demonstrate the effectiveness and superiority of the proposed framework.

This work addresses the lack of a unified computational framework for analyzing non-ergodicity, modeling heavy-tailed dynamics, and studying decision-making under uncertainty in stochastic processes. To this end, we introduce an open-source Python library that, for the first time, integrates non-ergodicity diagnostics, simulation of heavy-tailed processes—such as multiplicative Lévy growth and memory-dependent mean-reverting dynamics—and agent-based experimentation within a single platform. Built upon the scientific Python ecosystem (NumPy/SciPy), the library supports end-to-end workflows including stochastic process definition, simulation, parameter inference, and partial solution of stochastic differential equations. Through several reproducible examples—ranging from heavy-tailed ensemble diffusion to pre-asymptotic fluctuation analysis—it substantially reduces boilerplate code and enhances both reproducibility and development efficiency in the study of time-averaged behaviors of complex stochastic systems.

This work addresses the finite-time convergence of stochastic iterative algorithms for fixed-point equations accessible only through a noisy oracle. The authors propose a norm-independent, unified Lyapunov function framework constructed via a generalized Moreau envelope, which integrates Lyapunov stability theory with stochastic approximation analysis. This framework accommodates complex settings such as Markovian noise, seminorm contractive operators, and dissipative operators, yielding sharp non-asymptotic convergence bounds in both high-probability and mean-square senses. As a result, it provides a unified and refined finite-time convergence guarantee for a broad class of algorithms, including stochastic gradient descent, linear stochastic approximation, Q-learning, and temporal difference learning.

This work addresses a fundamental limitation of conventional reinforcement learning, which optimizes expected cumulative reward but fails to accurately capture the long-term performance of individual infinite-horizon trajectories in non-ergodic reward environments. Through theoretical examples, the paper systematically analyzes the impact of non-ergodic reward processes and demonstrates the inadequacy of standard objectives in reflecting real-world agent behavior under deployment. Leveraging ergodic Markov chain theory, the study reframes the evaluation of agent performance around individual trajectory outcomes rather than ensemble averages. Building on this insight, the authors synthesize and unify existing approaches into a coherent optimization framework explicitly designed to guarantee robust long-term performance along single trajectories. This framework provides both theoretical grounding and practical guidance for designing reinforcement learning objectives tailored to non-ergodic settings.