A quantitative “fourth moment theorem” for any self-adjoint element in a homogeneous Wigner chaos is provided: 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 analogues of the WS inequality and the WSH inequality are established.

中文翻译：

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自由概率理论中的定量第四矩定理

提供了均匀Wigner混沌中任何自伴元素的定量“第四矩定理”：Wasserstein距离由第四矩到二的距离控制。证明使用了斯坦因差异的免费副本。在此过程中，建立了WS不等式和WSH不等式的免费类似物。