We develop Stein’s method for \(\alpha \)-stable approximation with \(\alpha \in (0,1]\), continuing the recent line of research by Xu (Ann Appl Probab 29(1):458–504, 2019) and Chen et al. (J Theor Probab, 2018. https://doi.org/10.1007/s10959-020-01004-1) in the case \(\alpha \in (1,2)\). The main results include an intrinsic upper bound for the error of the approximation in a variant of Wasserstein distance that involves the characterizing differential operators for stable distributions and an application to the generalized central limit theorem. Due to the lack of first moment for the approximating sequence in the latter result, the proof strategy is significantly different from that in the integrable case. We rely on fine regularity estimates of the solution to Stein’s equation established in this paper.

我们继续用Xu（Ann Appl Probab 29（1）：458–504，1989）的最新研究成果，开发了用\（\ alpha \ in（0,1] \）稳定\（\ alpha \） -稳定近似的Stein方法。2019）和Chen等人（J Theor Probab，2018. https://doi.org/10.1007/s10959-020-01004-1）在\（\ alpha \ in（1,2）\）的情况下。主要结果包括Wasserstein距离变体中逼近误差的内在上限，其中包括表征稳定分布的微分算子以及对广义中心极限定理的应用。由于后一种结果中的近似序列缺少第一矩，因此证明策略与可积情况下的证明策略显着不同。我们依靠本文建立的Stein方程解的精细正则估计。