# On the rate of convergence in the central limit theorem for hierarchical Laplacian

## TL;DR

This paper investigates the rate at which the distribution of normalized eigenvalue means of a perturbed hierarchical Laplacian on ultrametric spaces converges to a normal distribution, providing estimates in total variation distance.

## Contribution

It extends previous CLT results for hierarchical Laplacians by quantifying the convergence rate in total variation distance under mild assumptions.

## Key findings

- Normalized eigenvalue means converge in law to a normal distribution.
- Provides explicit estimates for the rate of convergence in total variation distance.
- Enhances understanding of spectral properties of perturbed hierarchical Laplacians.

## Abstract

Let $(X,d)$ be a proper ultrametric space. Given a measure $m$ on $X$ and a function $C(B)$ defined on the set of all non-singleton balls $B$ we consider the hierarchical Laplacian $L=L_{C}$. Choosing a sequence $\{\varepsilon (B)\}$ of i.i.d. random variables we define the perturbed function $C(B,\omega )$ and the perturbed hierarchical Laplacian $L^{\omega }=L_{C(\omega )}.$ We study the arithmetic means $\overline{\lambda }(\omega )$ of the $L^{\omega }$-eigenvalues. Under some mild assumptions the normalized arithmetic means $\big( \overline{\lambda }-\mathbb{E}\overline{\lambda }\big) /\sigma \big( \overline{\lambda }\big) $ converge in law to the standard normal distribution. In this note we study convergence in the total variation distance and estimate the rate of convergence.