Non-integrable stable approximation by Stein's method

Abstract

We develop Stein's method for α-stable approximation with α ∈ (0, 1], continuing the recent line of research by Xu [40] and Chen, Nourdin and Xu [11] in the case 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, we appeal to an additional truncation procedure and investigate fine regularity properties of the solution to Stein's equation.

Multivariate stable approximation in Wasserstein distance by Stein's method

“…see [9,Section 2.3]. For the second equality in distribution we have used the self-similarity of the process (Z t ) t≥0 , namely Z ct = c 1/α Z t in distribution for any c, t > 0.…”

“…The approach of [21] was then generalized in [8] to asymmetric α-stable law with α > 1, and in [2] to a class of infinitely divisible distributions with finite first moment. Later, Chen et al [9] considered non-integrable α-stable approximation (necessarily α ≤ 1). In higher dimension, Arras and Houdré [3] carried out the aforementioned second step (construction of the solution to Stein's equation) for a class of self-decomposable distributions which includes multivariate stable laws.…”