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

• 成果类型: Article
• 署名作者: Borkovec, M; Klüppelberg, C
• 署名单位: Technical University of Munich
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• 发表日期: 2001
• 页码: 1220-1241
• 关键词: extremal behavior; markov-chains

摘要

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.