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Non-integrable Stable Approximation by Stein’s Method

Author

Listed:
  • Peng Chen

    (University of Macau)

  • Ivan Nourdin

    (Université du Luxembourg)

  • Lihu Xu

    (University of Macau)

  • Xiaochuan Yang

    (Université du Luxembourg
    National University of Singapore)

  • Rui Zhang

    (Capital Normal University)

Abstract

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.

Suggested Citation

  • Peng Chen & Ivan Nourdin & Lihu Xu & Xiaochuan Yang & Rui Zhang, 2022. "Non-integrable Stable Approximation by Stein’s Method," Journal of Theoretical Probability, Springer, vol. 35(2), pages 1137-1186, June.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:2:d:10.1007_s10959-021-01094-5
    DOI: 10.1007/s10959-021-01094-5
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    References listed on IDEAS

    as
    1. Kusuoka, Seiichiro & Tudor, Ciprian A., 2012. "Stein’s method for invariant measures of diffusions via Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1627-1651.
    2. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors," Papers 1212.6906, arXiv.org, revised Jan 2018.
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