## Asymptotic behaviour of the sample autocovariance and autocorrelation function of the $$AR(1)$$ process with $$\text{ARCH}(1)$$ errors.(English)Zbl 0996.62077

Summary: We consider a stationary AR(1) process with ARCH(1) errors given by the stochastic difference equation $X_t= \alpha X_{t-1}+\sqrt{\beta+ \lambda X_{t-1}^2} \varepsilon_t, \quad t\in \mathbb{N},$ where the $$(\varepsilon_t)$$ are independent and identically distributed symmetric random variables. In contrast to ARCH and GARCH processes, AR(1) processes with ARCH(1) errors are not solutions of linear stochastic recurrence equations and there is no obvious way to embed them into such equations. However, we show that they still belong to the class of stationary sequences with regularly varying finite-dimensional distributions and therefore the theory of R.A. Davis and T. Mikosch [Ann. Stat. 26, No. 5, 2049-2080 (1998; Zbl 0929.62092)] can be applied.
We present a complete analysis of the weak limit behaviour of the sample autocovariance and autocorrelation functions of $$(X_t)$$, $$(|X_t|)$$ and $$(X_t^2)$$. The results in this paper can be seen as a natural extension of results for ARCH(1) processes.

### MSC:

 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 60F05 Central limit and other weak theorems

Zbl 0929.62092
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