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.