Normal approximation for hierarchical structures

Abstract

Given F:[a,b]^k→[a,b] and a nonconstant X₀ with P(X₀∈[a,b])=1, define the hierarchical sequence of random variables {X_n}_n≥0 by X_n+1=F(X_n,1,…,X_n,k), where X_n,i are i.i.d. as X_n. Such sequences arise from hierarchical structures which have been extensively studied in the physics literature to model, for example, the conductivity of a random medium. Under an averaging and smoothness condition on nontrivial F, an upper bound of the form Cγⁿ for 0<γ<1 is obtained on the Wasserstein distance between the standardized distribution of X_n and the normal. The results apply, for instance, to random resistor networks and, introducing the notion of strict averaging, to hierarchical sequences generated by certain compositions. As an illustration, upper bounds on the rate of convergence to the normal are derived for the hierarchical sequence generated by the weighted diamond lattice which is shown to exhibit a full range of convergence rate behavior.