The Annals of Applied Probability

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

Milan Borkovec and Claudia Klüppelberg

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W consider the class of autoregressive processes with ARCH(1)errors given by the stochastic difference equation

$$X_n = \alpha X_{n-1} + \sqrt{\beta + \lambda X_{n-1}^2}\varepsilon_n,\quad n \in \mathbb{N}$$

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 $\alpha, \lambda$ and the distribution of the innovations $(\varepsilon_n)_{n \in \mathbb{N}}$. This paper generalizes results for the ARCH(1) process (the case $\alpha = 0$). The generalization requires a new method of proof and we invoke a Tauberian theorem.

Article information

Ann. Appl. Probab., Volume 11, Number 4 (2001), 1220-1241.

First available in Project Euclid: 5 March 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H25: Random operators and equations [See also 47B80]
Secondary: 60G10: Stationary processes 60J05: Discrete-time Markov processes on general state spaces

ARCH model autoregressive process geometric ergodicity heavy tail heteroscedastic model Markov process recurrent Harris chain regular variation Tauberian theorem


Borkovec, Milan; Klüppelberg, Claudia. The Tail of the Stationary Distribution of an Autoregressive Process with Arch(1) Errors. Ann. Appl. Probab. 11 (2001), no. 4, 1220--1241. doi:10.1214/aoap/1015345401.

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