Non-commutative Law of iterated logarithm
Operator Algebras
2025-09-29 v1 Functional Analysis
Probability
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.
Keywords
Cite
@article{arxiv.2509.22037,
title = {Non-commutative Law of iterated logarithm},
author = {Sourav Panja and Éric Ricard and Diptesh Saha},
journal= {arXiv preprint arXiv:2509.22037},
year = {2025}
}
Comments
16 pages