Central Limit Theorems for Nonlinear Hierarchical Sequences of Random Variables

Let a random variable x₀ and a function f:[a, b]^k[a, b] be given. A hierarchical sequence {x_n:n=0, 1, 2,...} of random variables is defined inductively by the relation x_n=f(x_{n–1, 1}, x_{n–1, 2}..., x_{n–1, k}), where {x_{n–1, i}:i=1, 2,..., k} is a family of independent random variables with the same distribution as x_n–1. We prove a central limit theorem for this hierarchical sequence of random variables when a function f satisfies a certain averaging condition. As a corollary under a natural assumption we prove a central limit theorem for a suitably normalized sequence of conductivities of a random resistor network on a hierarchical lattice.