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

Lorick Huang, Laurent Decreusefond, Laure Coutin

Abstract

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 infinite dimension. We show that provided we have a control between the randomwalk or the limiting stable process and their respective affine interpolation, we canlift the rate of convergence obtained for multivariate distributions to a rateof convergence in some functional spaces.

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

Abstract

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 infinite dimension. We show that provided we have a control between the randomwalk or the limiting stable process and their respective affine interpolation, we canlift the rate of convergence obtained for multivariate distributions to a rateof convergence in some functional spaces.
Paper Structure (14 sections, 12 theorems, 149 equations)

This paper contains 14 sections, 12 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Motivations
  3. Main result
  4. The Interpolated Random Walk, and Reduction to Finite Dimension
  5. Notations and the Interpolated Random Walk
  6. Reduction to finite dimension
  7. Control in Finite Dimension
  8. Approximation and Final Estimate
  9. Approximation Lemma
  10. The Interpolated Process Fits the setting of Lemma \ref{['ctrl:norme:holder']}
  11. Derivation of the Final Rate of convergence
  12. Proof of the Moment Condition
  13. The case of Symmetrized Pareto
  14. In the Normal domain of attraction

Key Result

Theorem 1

Let $(Y_{i})_{i\ge 1}$ be a sequence of IID random variables of cdf $F_{Y}$, in the normal domain of attraction of an $\alpha$-stable distribution with $\alpha\in (1,2)$. We assume that their exists $A>0,$$\gamma>0$ and a function $\varepsilon$ such that where $\varepsilon$ is a bounded function on $[-1,1]$ such that $\sup_{t}|t^{\gamma}\varepsilon(t)| <+\infty.$ Let $X_{n}$ be defined as in eq_T

Theorems & Definitions (27)

  • Theorem 1
  • Lemma 2
  • proof
  • Remark 1
  • Remark 2
  • Theorem 3
  • Corollary 4
  • Lemma 5
  • Remark 3
  • Remark 4
  • ...and 17 more