Strong approximation of the empirical process of GARCH sequences
The authors deal with a GARCH\((p,q)\) process defined by the equations \[ y_k=\sigma_k\varepsilon_k \quad\text{and}\quad \sigma_k^2 =\delta +\sum_{1\leq i\leq p} \beta_i \sigma_{k-i}^2 + \sum_{1\leq j\leq q} \alpha_j y_{k-j}^2 \] where \(\delta\), \(\beta_i\), and \(\alpha_i\) are nonnegative constants and \(\varepsilon_i\) are iid random variables. Let \(F\) be the distribution function of \(y_0\). Define the empirical process \(R(s,t)=\sum_{1\leq i\leq t} (I\{y_i\leq s\} -F(s))\). Under the assumption that there exists a unique stationary solution \(y_k\), a strong approximation for \(R(s,t)\) is obtained. This result implies the weak convergence of the empirical process and the law of the iterated logarithm.
scientific article; zbMATH DE number 1906240