In this note we extend the standard results for the limit theory of the popular quasi-maximum likelihood estimator (QMLE) in the context of the nonstationary autoregressive conditional heteroskedastic ARCH(1) model by allowing the innovation process not to possess fourth moments. Depending on the value of the index of stability, we either derive α-stable weak limits with nonstandard rates or inconsistency and non-tightness. We obtain the limit theory by the derivation of a limit theorem for multiplicative “martingale” transforms with limit mixtures of α-stable distributions for any α 2 0; 2 ð .

中文翻译：

---

关于非平稳ARCH（1）模型中QMLE极限理论的注记

在本注释中，我们通过允许创新过程不具有第四矩，在非平稳自回归条件异方差ARCH（1）模型的背景下扩展了流行的拟最大似然估计（QMLE）极限理论的标准结果。根据稳定性指数的值，我们可以得出具有非标准比率或不一致和不紧密的α稳定的弱极限。通过对任何α2 0具有α稳定分布的极限混合的乘法“ martingale”变换的极限定理的推导，我们得到了极限理论。2ð。