Empirical concentration for chains with infinite memory
Abstract
Concentration inequalities for time series usually involve unknown quantities measuring dependence. This prevents their direct use as confidence bounds, or as criteria for learning algorithms. Recent works proved empirical concentration inequalities for stationary Markov chains, where the unknown quantities are estimated. We extend this approach beyond the Markov case, by considering stationary chains with infinite memory. We first establish a bounded-differences inequality on bounded Polish metric spaces under a summable Wasserstein contraction condition. We then estimate its dependence coefficients from a single trajectory, using either transition counts when the state space is finite, or derivatives of kernel conditional-density estimates for a continuous state space. We give general error bounds and illustrate their dependence on the memory tail. Finally, a fixed multiplicative safety factor turns these estimators into empirical concentration bounds. While some regularity assumptions are required to make the bounds empirical, our procedure remains fully adaptive in the sense that it does not require the knowledge of the constants involved in these assumptions. The empirical confidence bounds derived from our results are asymptotically as tight as the theoretical ones.
Related
- Concentration of empirical entropy and transfer entropy for non-regular chains with unbounded memory
- Minimax estimation of the pseudo-spectral gap of continuous Markov chains, with applications to empirical concentration bounds
- Alternating Divisoriality of Prime Ideals along Infinite Chains of Bézout Domains
- Infinite chains of perfect fits for expanding Thurston maps
- Energy Decay Rate for a Porous-Elastic System with Fractional Infinite Memory
BibTeX
@misc{alquier2026empirical,
title = {{Empirical concentration for chains with infinite memory}},
author = {Pierre Alquier and Paul Doukhan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32265},
eprint = {2610.32265},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32265},
}
RIS
TY - UNPB
TI - Empirical concentration for chains with infinite memory
AU - Pierre Alquier
AU - Paul Doukhan
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32265
UR - https://arxiv.org/abs/2610.32265
AB - Concentration inequalities for time series usually involve unknown quantities measuring dependence. This prevents their direct use as confidence bounds, or as criteria for learning algorithms. Recent works proved empirical concentration inequalities for stationary Markov chains, where the unknown quantities are estimated. We extend this approach beyond the Markov case, by considering stationary chains with infinite memory. We first establish a bounded-differences inequality on bounded Polish metric spaces under a summable Wasserstein contraction condition. We then estimate its dependence coefficients from a single trajectory, using either transition counts when the state space is finite, or derivatives of kernel conditional-density estimates for a continuous state space. We give general error bounds and illustrate their dependence on the memory tail. Finally, a fixed multiplicative safety factor turns these estimators into empirical concentration bounds. While some regularity assumptions are required to make the bounds empirical, our procedure remains fully adaptive in the sense that it does not require the knowledge of the constants involved in these assumptions. The empirical confidence bounds derived from our results are asymptotically as tight as the theoretical ones.
ER -