Rate of Convergence in the Functional Central Limit Theorem for Stable Processes

Lorick Huang (1) , Laurent Decreusefond (2, 3) , Laure Coutin (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 RMS - Réseaux, Mobilité et Services
3 INFRES - Département Informatique et Réseaux

In this article, we quantify the functional convergence of the rescaled random walk with heavy tails to a stable process. This generalizes the Generalized Central Limit Theorem for stable random variables in finite dimension. We show that provided we have a control between the random walk or the limiting stable process and their respective affine interpolation, we can lift the rate of convergence obtained for multivariate distributions to a rate of convergence in some functional spaces.

Keywords: functional convergence Stein's method stable distribution central limit theorem; functional convergence; Stein's method; stable distribution; central limit theorem

Domains: Probability [math.PR]

Lorick Huang, Laurent Decreusefond, Laure Coutin. Rate of Convergence in the Functional Central Limit Theorem for Stable Processes. 2024. ⟨hal-04424279⟩