A law of large numbers and central limit theorem for the logarithm of an autoregressive process with a stationary driving sequence

Let (xi, eta, v) = {(xi(i), eta i(,) v(i))}(i is an element of z) be a stationary and ergodic sequence in R-3. Consider the autoregressive process defined by, R-0(xi, eta, v) = eta(0) and R-n(xi, eta, v) = xi R-n(n-1) (xi, eta, v) + eta(n)v(n), ... , v(1), n >= 1. We give conditions under which lim(n ->infinity) (1/n) In |R-n(xi, eta, v)| exists a.s. and at the same time, E In |xi(0)| < E In |v(0)|. We also generalize a Central Limit Theorem of Szekely for this process. (C) 2009 Published by Elsevier B.V.