Quantitative Estimates for Operator-Valued and Infinitesimal Boolean and Monotone Limit Theorems

TL;DR

This paper establishes quantitative bounds for operator-valued Boolean and monotone limit theorems, including CLTs and a fourth moment theorem, using a Lindeberg method for various types of independence.

Contribution

It introduces new Berry-Esseen bounds and a Lindeberg approach for operator-valued Boolean and monotone independence, extending to infinitesimal free, Boolean, and monotone CLTs.

Findings

Provides Berry-Esseen bounds in terms of first moments.

Quantitative bounds for operator-valued CLTs.

First estimates for infinitesimal free, Boolean, and monotone CLTs.

Abstract

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 L\'evy 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.