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arXiv Statistics ML By Pierre Alquier, Paul Doukhan

Empirical concentration for chains with infinite memory

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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.

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