We consider a class of perpetuities which admit direct characterization of asymptotics of the key truncated moment. The class contains perpetuities without polynomial decay of tail probabilities and thus not satisfying Kesten's theorem. We show how to apply this result in deriving a new weak law of large numbers for solutions to stochastic recurrence equations and a new central limit theorem for GARCH(1,1) processes in the critical case.

中文翻译：

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GARCH 过程的截断矩和新的中心极限定理没有 Kesten 的规律性

我们考虑一类允许对关键截断矩的渐近线进行直接表征的永续性。该类包含没有尾概率多项式衰减的永续性，因此不满足 Kesten 定理。我们展示了如何应用这个结果来推导出一个新的弱大数定律来解决随机递推方程的解和一个新的 GARCH(1,1) 过程在临界情况下的中心极限定理。