The paper presents empirical concentration inequalities for stationary chains with infinite memory, extending previous results that were limited to Markov chains. It establishes a bounded‑differences inequality under a summable Wasserstein contraction condition and shows how to estimate the necessary dependence coefficients from a single trajectory, using transition counts for finite state spaces or kernel density derivatives for continuous spaces. The authors provide general error bounds that reflect the memory tail and demonstrate that, after applying a fixed safety factor, the resulting empirical bounds are asymptotically as tight as the theoretical ones, while remaining fully adaptive without requiring knowledge of the underlying constants.
arXiv:2610.20007v1 Announce Type: cross Abstract: We study the minimax estimation of the pseudo-spectral gap $\gamma_{ps}$ of a Markov chain from the observation of a single trajectory. Extending ear...
arXiv:2607. 22399v1 Announce Type: cross Abstract: We consider the problem of learning from a single finite trajectory of an ergodic stochastic dynamical system.
arXiv:2607. 17595v1 Announce Type: new Abstract: We establish mean-square and concentration bounds for stochastic approximation (SA) with arbitrary norm contractive mappings, under a multiplicative noise model where the noise may scale affinely with the norm of the iterates, and the iterates are potentially unbounded.
The paper investigates how Markov chains converge within a geometry defined by Hilbertian embeddings of positive definite kernels, also known as Maximum Mean Discrepancy (MMD). By assuming a contraction in this kernel-induced geometry and a standard spectral gap, the authors derive non-asymptotic, high‑probability bounds on the empirical occupation measure’s convergence to the invariant measure. These bounds separate the influence of the spectral gap, which sets the effective sample size, from the kernel contraction, which appears only as a linear correction. The results are applied to one‑step prediction of dynamical systems from a single trajectory, yielding fast learning rates for a nonlinear least‑squares estimator that illustrate the interaction between kernel‑induced mixing and classical source and capacity conditions.
arXiv:2609.38524v1 Announce Type: new Abstract: We consider estimating the one-step-ahead conditional distribution of a multivariate stochastic process. Many existing approaches rely on assumptions s...
arXiv:2602.13960v2 Announce Type: replace Abstract: Constant-stepsize stochastic approximation (SA) is widely used in learning for computational efficiency, yet the distribution of the iterates is ty...
arXiv:2610.06894v1 Announce Type: cross Abstract: A central question in the prediction of stochastic processes is the extent to which past information can improve the probability of correctly predict...
Suppose we observe the first $n$ points of a sequence of random variables having length $n+1$, and wish to estimate a functional of the unobserved final point and the empirical measure of the $n$ obse...
The paper introduces a leave‑a‑window‑out estimator for next‑token functionals, such as the surprise probability and test error, in sequences of random variables. By deleting a window of length τ after each index, the estimator generalizes leave‑one‑out and achieves parametric error decay for stationary β‑mixing processes that admit a Marton coupling. The authors provide both upper bounds and a minimax lower bound for the surprise probability, and demonstrate through simulations that their method outperforms traditional baselines on Markov, moving‑average, and autoregressive processes.
arXiv:2606. 19117v1 Announce Type: cross Abstract: Offline policy learning has received growing attention in causal inference.
arXiv:2608. 14408v1 Announce Type: cross Abstract: We study online statistical inference for functionals of the return distribution under a fixed policy.
The paper establishes a shrinking‑tube concentration bound for projected stochastic approximation driven by an adaptive Markov chain, guaranteeing that after a chosen time every iterate stays within a tolerance that tightens over time. The bound’s probability of any exit after that time decays polynomially, and a matching lower bound shows this exponent is optimal under finite second moments. Extensions to recursions with martingale‑difference noise and predictable bias reveal how noise scale and bias affect exit‑probability decay and tube shrinkage, with applications to inventory learning and numerical gradient accuracy.