A central limit theorem in the framework of the Thompson group $F$

TL;DR

This paper proves a central limit theorem for a sequence of elements in the group algebra of the Thompson group F, showing that their normalized sum converges to a standard normal distribution.

Contribution

It establishes a new central limit theorem within the noncommutative probability framework of the Thompson group F, linking group algebra elements to classical Gaussian limits.

Findings

The sequence of normalized sums converges to the standard normal distribution.

The result applies to the generators of the Thompson group F in its standard presentation.

The theorem extends classical CLT concepts to a noncommutative group algebra setting.

Abstract

We discuss a central limit theorem in the framework of the group algebra of the Thompson group $F$. We consider the sequence of self-adjoint elements given by $a_n=\frac{g_n+g_n^{*}}{\sqrt{2}}$ in the noncommutative probability space $(\mathbb{C}(F),\varphi)$, where the expectation functional $\varphi$ is the trace associated to the left regular representation of $F$, and the $g_n$-s are the generators of $F$ in its standard infinite presentation. We show that the limit law of the sequence $s_n = \frac{a_0+\cdots+a_{n-1}}{\sqrt{n}}$ is the standard normal distribution.