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

Let (X, d) be a proper ultrametric space. Given a measure m on X and a function B -> C B) defined on the set of all non-singleton balls B we consider the hierarchical Laplacian L = L-C. Choosing a sequence {epsilon( 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 (lambda) over bar(omega) of the L-omega-eigenvalues. Under certain assumptions the normalized arithmetic means ((lambda) over bar -E (lambda) over bar)/sigma((lambda) over bar) 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.