论文标题
自由熵和Stein差异的最大相关性和单调性
Maximal correlation and monotonicity of free entropy and Stein discrepancy
论文作者
论文摘要
我们介绍了两个非共同概率子空间$ m_1 $和$ m_2 $之间的最大相关系数$ r（m_1，m_2）$，并表明由$ s_n产生的子代数之间的最大相关系数$ \ sqrt {m/n} $ for $ m \ le n $，其中$（x_i）_ {i \ in \ mathbb {n}} $是一系列免费且相同分布的非交易性随机变量。这是Dembo-Kagan--Shepp在经典概率中的结果的自由概率类似物。作为应用程序，我们使用此估算值提供了另一个简单的证据，证明了自由熵的单调性和在自由中央限制定理中的自由渔民信息。此外，我们证明了Fathi和Nelson引入的自由Stein差异沿着自由中央限制定理不断增加。
We introduce the maximal correlation coefficient $R(M_1,M_2)$ between two noncommutative probability subspaces $M_1$ and $M_2$ and show that the maximal correlation coefficient between the sub-algebras generated by $s_n:=x_1+\ldots +x_n$ and $s_m:=x_1+\ldots +x_m$ equals $\sqrt{m/n}$ for $m\le n$, where $(x_i)_{i\in \mathbb{N}}$ is a sequence of free and identically distributed noncommutative random variables. This is the free-probability analogue of a result by Dembo--Kagan--Shepp in classical probability. As an application, we use this estimate to provide another simple proof of the monotonicity of the free entropy and free Fisher information in the free central limit theorem. Moreover, we prove that the free Stein Discrepancy introduced by Fathi and Nelson is non-increasing along the free central limit theorem.
