We study the effect of observing a stationary process at irregular time points via a renewal process. We establish a sharp difference in the asymptotic behaviour of the self-normalized sample mean of the observed process depending on the renewal process. In particular, we show that if the renewal process has a moderate heavy tail distribution then the limit is a so-called Normal Variance Mixture (NVM) and we characterize the randomized variance part of the limiting NVM as an integral function of a L\'evy stable motion. Otherwise, the normalized sample mean will be asymptotically normal.

翻译：本文研究通过更新过程在非规则时间点观测平稳过程的影响。我们建立了观测过程自标准化样本均值渐近行为的显著差异，该差异取决于更新过程的特性。具体而言，我们证明若更新过程具有中等重尾分布，则极限分布为所谓的正态方差混合分布，并将该极限分布中随机化方差部分刻画为Lévy稳定运动的积分函数。反之，标准化样本均值将具有渐近正态性。