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

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.