## 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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