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

¹ Instytut Matematyczny, Uniwersytet Wrocławski, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
² Institut für Diskrete Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria.

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 24 February 2017
Accepted: 13 March 2018

Abstract

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 {ε(B)} of i.i.d. random variables we define the perturbed function C(B, ω) and the perturbed hierarchical Laplacian L^ω = L_C(ω). We study the arithmetic means λ^̅(ω) of the L^ω-eigenvalues. Under certain assumptions the normalized arithmetic means (λ^̅−Eλ^̅) ∕ σ(λ^̅) 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.