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Multivariate CLT for Lévy processes: convergence rates without moment assumptions

Jorge González Cázares, David Kramer-Bang, Aleksandar Mijatović

TL;DR

The paper establishes a complete, dimension-aware characterization of CLT convergence speeds for $d$-dimensional Lévy processes without moment assumptions. It proves that finite second moments are equivalent to the local integrability of scale-invariant convex (and Kolmogorov) distances to a Gaussian under appropriate centering and scaling, and that the classical $\sqrt{t}$-scaling yields integrability precisely when $\mathbb{E}[|\bm X_1|^2\max\{0,\log|\bm X_1|\}]<\infty$ (a $(2+\log)$-moment). The results hold in the genuinely multivariate setting and relate to the domain of attraction classifications, providing hard limits on polynomial Berry-Esseen-type bounds in higher dimensions. The authors develop a novel multivariate extension of small-time limit theorems and combine Berry-Esseen techniques for truncated jumps with matrix-interpolation arguments to derive these equivalences, and they discuss implications for discrete-time analogues and Wasserstein-type distances. Overall, the work clarifies fundamental limits on convergence rates in multivariate CLTs for Lévy processes and connects moment conditions to sharp integrability properties of convergence metrics.

Abstract

We prove that the norm of a $d$-dimensional Lévy process possesses a finite second moment if and only if the convex distance between an appropriately rescaled process at time $t$ and a standard Gaussian vector is integrable in time with respect to the scale-invariant measure $t^{-1} dt$ on $[1,\infty)$. We further prove that under the standard $\sqrt{t}$-scaling, the corresponding convex distance is integrable if and only if the norm of the Lévy process has a finite $(2+\log)$-moment. Both equivalences also hold for the integrability with respect to $t^{-1} dt$ of the multivariate Kolmogorov distance. Our results imply: (I) polynomial Berry-Esseen bounds on the rate of convergence in the convex distance in the CLT for Lévy processes cannot hold without finiteness of $(2+δ)$-moments for some $δ>0$ and (II) integrability of the convex distance with respect to $t^{-1} dt$ in the domain of non-normal attraction cannot occur for any scaling function.

Multivariate CLT for Lévy processes: convergence rates without moment assumptions

TL;DR

The paper establishes a complete, dimension-aware characterization of CLT convergence speeds for -dimensional Lévy processes without moment assumptions. It proves that finite second moments are equivalent to the local integrability of scale-invariant convex (and Kolmogorov) distances to a Gaussian under appropriate centering and scaling, and that the classical -scaling yields integrability precisely when (a -moment). The results hold in the genuinely multivariate setting and relate to the domain of attraction classifications, providing hard limits on polynomial Berry-Esseen-type bounds in higher dimensions. The authors develop a novel multivariate extension of small-time limit theorems and combine Berry-Esseen techniques for truncated jumps with matrix-interpolation arguments to derive these equivalences, and they discuss implications for discrete-time analogues and Wasserstein-type distances. Overall, the work clarifies fundamental limits on convergence rates in multivariate CLTs for Lévy processes and connects moment conditions to sharp integrability properties of convergence metrics.

Abstract

We prove that the norm of a -dimensional Lévy process possesses a finite second moment if and only if the convex distance between an appropriately rescaled process at time and a standard Gaussian vector is integrable in time with respect to the scale-invariant measure on . We further prove that under the standard -scaling, the corresponding convex distance is integrable if and only if the norm of the Lévy process has a finite -moment. Both equivalences also hold for the integrability with respect to of the multivariate Kolmogorov distance. Our results imply: (I) polynomial Berry-Esseen bounds on the rate of convergence in the convex distance in the CLT for Lévy processes cannot hold without finiteness of -moments for some and (II) integrability of the convex distance with respect to in the domain of non-normal attraction cannot occur for any scaling function.

Paper Structure

This paper contains 13 sections, 21 theorems, 130 equations.

Table of Contents

  1. Introduction and main results
  2. Question I
  3. The variance is finite if and only if the scaled distance d_C is integrable at infinity
  4. Question II
  5. Integrability of the scaled distance in the CLT is equivalent to (2+log)-moments
  6. Main contributions and related literature
  7. Strategy for the proofs
  8. Proofs of Theorem \ref{['thm:int-2']} and of the implication (a) -> (b) in Theorem \ref{['thm:int-0']}
  9. Proof of the implication (b) implies (a) in Theorem \ref{['thm:int-0']}
  10. Concluding remarks
  11. From continuous to discrete time
  12. The Wasserstein distance
  13. Convergence metrics and the proof of the characterisation of the multidimensional domain of normal attraction

Key Result

Theorem 1.1

Let $\bm{X}$ be a genuinely $d$-dimensional Lévy process and $\bm{Z}$ a standard Gaussian random vector in $\mathbb{R}^d$. Let $\mathscr{A}$ be either $\mathscr{K}$ or $\mathscr{C}$ defined above. Then the following conditions are equivalent. Moreover, for any function $\bm{B}$ satisfying condition (b), we have $\lim_{t\to\infty}d_\mathscr{C}(\bm{X}_t-t\mathds{E}\bm{X}_1,\bm{B}(t) \bm{Z} )=0$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Proposition 2.2
  • proof : Proof of Proposition \ref{['prop:multi_extention_Asmussen']}
  • Lemma 2.3
  • proof
  • ...and 31 more