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The tail of the stationary distribution of an autoregressive process with \(\text{ARCH}(1)\) errors. (English) Zbl 1010.62083

Summary: We consider the class of autoregressive processes with ARCH(1) errors given by the stochastic difference equation \[ X_n=\alpha X_{n-1}+ \sqrt{\beta+ \lambda X^2_{n-1}} \varepsilon_n, \quad n\in\mathbb{N}, \] 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.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60G10 Stationary stochastic processes
60J05 Discrete-time Markov processes on general state spaces
62G32 Statistics of extreme values; tail inference
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