The Tail of the Stationary Distribution of an Autoregressive Process with Arch(1) Errors

W consider the class of autoregressive processes with ARCH(1)errors given by the stochastic difference equation Xn=.Xn.1+..+.X2n.1.n,n.N where (\varepsilon_n) {n \in \mathbb{N} are i.i.d random variables. Under general and tractable assumptions we show the existence and uniqueness of a stationary distribution. We prove that the stationary distribution has a Pareto-like tail with a well-specified tail index which depends on .,. and the distribution of the innovations (.n)n.N. This paper generalizes results for the ARCH(1) process (the case .=0). The generalization requires a new method of proof and we invoke a Tauberian theorem.