Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitative and stable limits of high-frequency statistics of Lévy processes: a Stein's method approach

Chiara Amorino, Arturo Jaramillo, Mark Podolskij

TL;DR

The paper develops a Stein's method framework to study high-frequency statistics of multidimensional Lévy processes and their convergence to mixed Gaussian limits. It proves a general stable-convergence criterion (Theorem ['stein']) to mixed normal limits, without embedding into a Gaussian space, and derives quantitative rates (Theorem ['th1']) for the convergence of the normalized statistic $Z^{(n)}$ to a Gaussian integral driven by a stochastic variance $A_t$. The results are organized around tail regimes indexed by $\alpha$ through hypotheses $m{H_1}(m{\alpha})$, with refined rates under stronger conditions ($\bm{H_2}$, $\bm{H_3}$). The methodology blends Stein's method with a $K$-function–style decomposition to handle stochastic variances and non-central limits, offering a general toolkit for stable limit theorems in non-Gaussian high-frequency settings and potential extensions to broader stochastic environments.

Abstract

We establish inequalities for assessing the distance between the distribution of errors of partially observed high-frequency statistics of multidimensional Lévy processes and that of a mixed Gaussian random variable. Furthermore, we provide a general result guaranteeing stable functional convergence. Our arguments rely on a suitable adaptation of the Stein's method perspective to the context of mixed Gaussian distributions, specifically tailored to the framework of high-frequency statistics.

Quantitative and stable limits of high-frequency statistics of Lévy processes: a Stein's method approach

TL;DR

The paper develops a Stein's method framework to study high-frequency statistics of multidimensional Lévy processes and their convergence to mixed Gaussian limits. It proves a general stable-convergence criterion (Theorem ['stein']) to mixed normal limits, without embedding into a Gaussian space, and derives quantitative rates (Theorem ['th1']) for the convergence of the normalized statistic to a Gaussian integral driven by a stochastic variance . The results are organized around tail regimes indexed by through hypotheses , with refined rates under stronger conditions (, ). The methodology blends Stein's method with a -function–style decomposition to handle stochastic variances and non-central limits, offering a general toolkit for stable limit theorems in non-Gaussian high-frequency settings and potential extensions to broader stochastic environments.

Abstract

We establish inequalities for assessing the distance between the distribution of errors of partially observed high-frequency statistics of multidimensional Lévy processes and that of a mixed Gaussian random variable. Furthermore, we provide a general result guaranteeing stable functional convergence. Our arguments rely on a suitable adaptation of the Stein's method perspective to the context of mixed Gaussian distributions, specifically tailored to the framework of high-frequency statistics.
Paper Structure (23 sections, 9 theorems, 198 equations)

This paper contains 23 sections, 9 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. The setting and overview
  3. Main results
  4. Proof of Theorem \ref{['stein']}
  5. Proof of Theorem \ref{['th1']}
  6. Stein's approach and some definitions
  7. The main decomposition
  8. Treatment of the term $R_{1,n}$
  9. Case $\alpha\in(0,2)$ under condition $\bf{H}_1(\alpha)$
  10. Case $\alpha\in(1,2]$ under condition $\bf{H}_2(\alpha)$
  11. Analysis of $T_{2,n}$
  12. Supplementary material
  13. Proof of Proposition \ref{['prop: T negl']}
  14. Proof under condition $\bf{H_1}(\alpha)$
  15. Analysis of $T_{1,n}$
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let $\bm{X}$ be a symmetric $\alpha$-stable process with $\alpha\in(0,2)$ and assume that $g$ is such that $\|g\|_{\infty},\|\frac{\partial g}{\partial x_{i}}\|_{\infty},\|\frac{\partial^2 g}{\partial x_{i}\partial x_{j}}\|_{\infty}<\infty,$ for all $i,j=1,\dots, d$. Then, the function is well defined. Under these conditions, for all $t>0$, there exists a standard Brownian motion $W$ independent

Theorems & Definitions (21)

  • Theorem 2.1: Special case of Theorem \ref{['th1']}
  • Remark 2.2
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • Remark 3.6
  • Theorem 3.7
  • Remark 3.8
  • ...and 11 more