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Self-normalized Bernstein Inequalities for Regenerative Processes and Applications to Learning Without Independence

Imon Banerjee, Pierre Alquier · 2026 · arXiv

DOI: 10.48550/arXiv.2610.30119 · open-access PDF

Japanese

We prove a self-normalized Bernstein inequality for bounded functions of a regenerative process with independent, identically distributed marked cycles of finite mean length. The bound uses observed cycle lengths and retains explicit terms for truncation and the cycle ending after observation horizon. Finite deterministic collection selecting a tuning parameter gives a version of the Bernstein inequality where all terms are empirical but one, related to the tails of the regeneration times. We then study this term in many regimes: it turns out that it is of second-order in many cases of regeneration times. 結局のところ、このサイクルは再生期の尾と関連している。

Machine translation (Meta M2M100); the English original follows.

Abstract

We prove a self-normalized Bernstein inequality for bounded functions of a regenerative process with independent, identically distributed marked cycles of finite mean length. The bound uses observed cycle lengths and retains explicit terms for truncation and the cycle ending after the observation horizon. Selecting a tuning parameter from a finite deterministic collection gives a version of Bernstein inequality where all terms are empirical but one, related to the tails of the regeneration times. We then study this term in many regimes: it turns out that it is of second-order in many cases of

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