Non-commutative Law of iterated logarithm

2025/09/26 by Panja, Sourav, Ricard, Éric, Saha, Diptesh
#46L53 #46L55 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Probability (math.PR)

paper · doi:10.48550/arxiv.2509.22037

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.

Citations

• Noncommutative martingale deviation and Poincaré type inequalities with applications
• Noncommutative maximal ergodic theorems
• Doob's inequality for non-commutative martingales
• On the Law of the Iterated Logarithm

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• On the best constants in some non-commutative martingale inequalities