The tail of the stationary distribution of an autoregressive process with ARCH(1) errors

We consider the class of autoregressive processes with ARCH(1) errors given by the stochastic difference equation X-n = alphaX(n-1) + rootbeta + lambdaX(n-1)(2) epsilon(n), nis an element of N where (epsilon(n))(nepsilonN) 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 Paretu-like tail with a well-specified tail index which depends on alpha, lambda and the distribution of the innovations This paper generalizes results for the ARCH(l) process (the case alpha = 0). The generalization requires a new method of proof and we invoke a Tauberian theorem.