A quantitative fourth moment theorem in free probability theory
Guillaume C\'ebron
TL;DR
This paper establishes a quantitative fourth moment theorem in free probability, linking Wasserstein distance to the fourth moment deviation, and introduces a free Stein discrepancy and a free WSH inequality.
Contribution
It provides the first quantitative fourth moment theorem in free probability and develops the free Stein discrepancy and free WSH inequality.
Findings
Wasserstein distance is controlled by the fourth moment deviation.
Established the free analogue of the WSH inequality.
Introduced a free Stein discrepancy for self-adjoint elements.
Abstract
A quantitative "fourth moment theorem" is provided for any self-adjoint element in a homogeneous Wigner chaos: the Wasserstein distance is controlled by the distance from the fourth moment to two. The proof uses the free counterpart of the Stein discrepancy. On the way, the free analogue of the WSH inequality is established.
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