Remarks on stationary GARCH processes under heavy tail distributions

Let $(X_n{)}_{n\in \mathbb{Z}}$ be a GARCH process with $E(X_0^4)<\mathrm{\infty }$, and let ${\mu }_n$ denote the distribution of $\frac{1}{\sqrt{n}}{\sum }_{i=1}^n[X_i^2-\mathbb{E}(X_0^2)]$. We derive a numerical approximation of ${\mu }_n$ when $x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n$ are observed. This yields the derivation of confidence intervals for $\mu =E(X_0^2)$ and we investigate the accuracy of these confidence intervals in comparison with standard ones based on normal approximation. Moreover, when the innovation process has heavy tail distribution, we improve the method using a new resampling method.