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November 2001 The Tail of the Stationary Distribution of an Autoregressive Process with Arch(1) Errors
Milan Borkovec, Claudia Klüppelberg
Ann. Appl. Probab. 11(4): 1220-1241 (November 2001). DOI: 10.1214/aoap/1015345401

Abstract

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.

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Milan Borkovec. Claudia Klüppelberg. "The Tail of the Stationary Distribution of an Autoregressive Process with Arch(1) Errors." Ann. Appl. Probab. 11 (4) 1220 - 1241, November 2001. https://doi.org/10.1214/aoap/1015345401

Information

Published: November 2001
First available in Project Euclid: 5 March 2002

zbMATH: 1010.62083
MathSciNet: MR1878296
Digital Object Identifier: 10.1214/aoap/1015345401

Subjects:
Primary: 60H25
Secondary: 60G10 , 60J05

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

Rights: Copyright © 2001 Institute of Mathematical Statistics

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Vol.11 • No. 4 • November 2001
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