Non-integrable Stable Approximation by Stein’s Method

We develop Stein’s method for $$\alpha $$ α -stable approximation with $$\alpha \in (0,1]$$ α ∈ ( 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 $$\alpha \in (1,2)$$ α ∈ ( 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.