论文标题
运算符值和无限布尔和单调定理的定量估计值
Quantitative Estimates for Operator-Valued and Infinitesimal Boolean and Monotone Limit Theorems
论文作者
论文摘要
我们提供浆果 - 埃森的界限，以根据汇总的第一瞬间来提供运营商价值布尔和单调自变量的总和。我们的边界位于凯奇变换的水平以及Lévy距离的水平。作为应用，我们获得了相应的CLT的定量界限，为单调独立的随机变量（包括操作员价值的情况）提供了定量的“第四钟定理”，并将Hao和Popa的结果推广到带有Boolean条目的矩阵上。我们的方法取决于我们为布尔/单调独立随机变量的总和而开发的lindeberg方法。此外，我们将这种方法推向无穷小设置，以获得对操作员可导出的无限无限无限，布尔和单调CLT的首次定量估计值。
We provide Berry-Esseen bounds for sums of operator-valued Boolean and monotone independent variables, in terms of the first moments of the summands. Our bounds are on the level of Cauchy transforms as well as the Lévy distance. As applications, we obtain quantitative bounds for the corresponding CLTs, provide a quantitative "fourth moment theorem" for monotone independent random variables including the operator-valued case, and generalize the results by Hao and Popa on matrices with Boolean entries. Our approach relies on a Lindeberg method that we develop for sums of Boolean/monotone independent random variables. Furthermore, we push this approach to the infinitesimal setting to obtain the first quantitative estimates for the operator-valued infinitesimal free, Boolean and monotone CLT.
