A Central Limit Theorem for Operators

TL;DR

This paper establishes a Central Limit Theorem analogue for operators, showing convergence of operator sequences to a unique centered Gaussian operator, and connects this to approximation methods and Hermite semigroups.

Contribution

It introduces a novel CLT for operators, characterizes the limiting operator as a Centered Gaussian Operator, and links Beckner's approximation method to a broader operator framework.

Findings

Operators converge to a unique Centered Gaussian Operator

The Hermite semi-group is characterized as a family of Centered Gaussian Operators

Beckner's approximation method is a special case of a general operator approximation

Abstract

We prove an analogue of the Central Limit Theorem for operators. For every operator $K$ defined on $\mathbb{C}[x]$ we construct a sequence of operators $K_N$ defined on $\mathbb{C}[x_1,...,x_N]$ and demonstrate that, under certain orthogonality conditions, this sequence converges in a weak sense to an unique operator $\mathcal{C}$. We show that this operator $\mathcal{C}$ is a member of a family of operators $\mathfrak{C}$ that we call {\it Centered Gaussian Operators} and which coincides with the family of operators given by a centered Gaussian Kernel. Inspired in the approximation method used by Beckner in [W. Beckner, Inequalities in Fourier Analysis, Annals of Mathematics, 102 (1975), 159-182] to prove the sharp form of the Hausdorff-Young inequality, the present article shows that Beckner's method is a special case of a general approximation method for operators. In particular, we characterize the Hermite semi-group as the family of Centered Gaussian Operators associated with any semi-group of operators.