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arXiv · 2509.22037

Non-commutative Law of iterated logarithm

Sourav Panja·Éric Ricard·Diptesh Saha
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Abstract

We prove optimal non-commutative analogues of the classical Law of Iterated Logarithm (LIL) for both martingales and sequences of independent (non-commutative) random variables. The classical martingale version was established by Stout [Sto70b] and the independent case by Hartman-Wintner [HW41]. Our approach relies on a key exponential inequality essentially due to Randrianantoanina [Ran24] that improves that from Junge and Zeng [JZ15]. It allows to derive an optimal non-commutative Stout-type LIL just as in [Zen15], from that martingale result we then deduce a non-commutative Hartman-Wintner type LIL for independent sequences of random variables.

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Sourav Panja, Éric Ricard, Diptesh Saha. 2025-09-26. Non-commutative Law of iterated logarithm. https://arxiv.org/abs/2509.22037

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