Let (X, d) be a locally compact separable ultrametric space. Given a measure m on X and a function C(B) defined on the set of all non-singleton balls B of X we consider the hierarchical Laplacian L = LC . The operator L acts in L2(X, m), is essentially self-adjoint and has a purely point spectrum. Choosing a family {ε(B)} of i.i.d. we define the perturbated function C(B, ω) and the perturbated hierarchical Laplacian Lω = LC(ω). We study the arithmetic means λ(ω) of the Lω-eigenvalues. Under some mild assumptions the normalized arithmetic means λ − Eλ /σλ converge to N(0, 1) in law. We also give examples where the normal convergence fails. We prove existence of the integrated density of states.