Central limit theorems for nonlinear hierarchical sequences of random variables

Let a random variable xo 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.