Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sharp total variation rates of convergence for fluctuations of linear statistics of $β$-ensembles

Jürgen Angst, Ronan Herry, Dominique Malicet, Guillaume Poly

Abstract

In this article, we revisit the question of fluctuations of linear statistics of beta ensembles in the single cut and non-critical regime for general potentials $V$ under mild regularity and growth assumptions. Our main objective is to establish sharp quantitative Central Limit Theorems (CLT) for strong distances, such as the total variation distance, which to the best of our knowledge, is new for general potentials, even qualitatively. Namely, setting $μ_V$ the equilibrium measure, for a test function $ξ\in \mathscr{C}^{14}$, we establish the convergence in total variation of $X_n=\sum_{i=1}^n ξ(λ_i)-n\langle ξ,μ_V\rangle$ to an explicit Gaussian variable at the sharp speed $1/n$. Under the same assumptions, we also establish multivariate CLTs for vectors of linear statistics in $p-$Wasserstein distances for any $p\ge 1$, with the optimal rate $1/n$, a result which already in dimension one sharpens the speed of convergence established in the recent contribution [26] as well as the required regularity on the test functions. A second objective of this paper, in a more qualitative direction, is to establish the so-called super-convergence of linear statistics, that is to say the convergence of all derivatives of the densities of $X_n$ uniformly on $\mathbb{R}$, provided that $ξ\in\mathscr{C}^\infty(\mathbb{R})$ and is not too degenerated in some sense.

Sharp total variation rates of convergence for fluctuations of linear statistics of $β$-ensembles

Abstract

In this article, we revisit the question of fluctuations of linear statistics of beta ensembles in the single cut and non-critical regime for general potentials under mild regularity and growth assumptions. Our main objective is to establish sharp quantitative Central Limit Theorems (CLT) for strong distances, such as the total variation distance, which to the best of our knowledge, is new for general potentials, even qualitatively. Namely, setting the equilibrium measure, for a test function , we establish the convergence in total variation of to an explicit Gaussian variable at the sharp speed . Under the same assumptions, we also establish multivariate CLTs for vectors of linear statistics in Wasserstein distances for any , with the optimal rate , a result which already in dimension one sharpens the speed of convergence established in the recent contribution [26] as well as the required regularity on the test functions. A second objective of this paper, in a more qualitative direction, is to establish the so-called super-convergence of linear statistics, that is to say the convergence of all derivatives of the densities of uniformly on , provided that and is not too degenerated in some sense.
Paper Structure (38 sections, 14 theorems, 112 equations)

This paper contains 38 sections, 14 theorems, 112 equations.

Table of Contents

  1. Introduction and statement of the results
  2. Overview of our contributions
  3. Motivations
  4. Sharp quantitative Central Limit Theorems
  5. Implementing a robust Stein's approach for β-ensembles
  6. Super-convergence phenomenon
  7. Statement of the main results
  8. Assumption and notations
  9. Quantitative CLT in total variation and Wasserstein distances
  10. Super-convergence
  11. Comparison with existing results
  12. Gaussian fluctuations for linear statistics
  13. Stein's method and β-ensemble
  14. Super-convergence
  15. Outline of the proofs an plan of the paper
  16. ...and 23 more sections

Key Result

Theorem 1.1

Let us fix $1 \leq p < \infty$ and $\alpha > 0$. There exists $K_{p,\alpha} > 0$ such that for all test functions $\xi_{1}, \dots, \xi_{d} \in \mathscr{C}^{{6}}(\mathbb{R})$ satisfying eq:free, we have, with $N$ a standard Gaussian in $\mathbb R^d$ In the univariate case $d=1$, with $\sigma:=\sqrt{c_{11}}$ and $m=m_1$, we also have

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Lemma 2.2: Stein kernel bounds LNPFathinourdin2009stein
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Remark 2.5
  • ...and 12 more