Multivariate Stable Approximation by Stein’s Method

Peng Chen (), Ivan Nourdin (), Lihu Xu () and Xiaochuan Yang ()
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Peng Chen: Nanjing University of Aeronautics and Astronautics
Ivan Nourdin: Université du Luxembourg
Lihu Xu: University of Macau
Xiaochuan Yang: Brunel University

Journal of Theoretical Probability, 2024, vol. 37, issue 1, 446-488

Abstract: Abstract By a delicate analysis for the Stein’s equation associated with the $$\alpha $$ α -stable law approximation with $$\alpha \in (0,2)$$ α ∈ ( 0 , 2 ) , we prove a quantitative stable central limit theorem in Wasserstein-type distance, which generalizes the results in the series of work (Chen et al. in J Theor Probab 34(3):1382–1407, 2021; Chen et al. in J Theor Probab 35(2):1137–1186 2022; Xu in Ann Appl Probab 29(1):458–504, 2019) from the univariate case to the multiple variate case. From an explicit computation for Pareto’s distribution, we see that the rate of our approximation is sharp. The analysis of the Stein’s equation is new and has independent interest.

Keywords: Multivariate $$\alpha $$ α -stable approximation; Stein’s method; Generalized central limit theorem; Rate of convergence; Wasserstein(-type) distance; Fractional Laplacian; 60E07; 60E17; 60F05; 60G52 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10959-023-01244-x

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