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Quantitative central limit theorem for an integrated periodogram via the fourth moment theorem

Samir Ben Hariz, Duc Quang Bui, Youssef Esstafa

Abstract

We revisit the central limit theorem for integrated periodograms, equivalently for Toeplitz quadratic forms of stationary Gaussian sequences. Under a regular-variation assumption allowing long-memory singularities and slowly varying corrections, we prove a quantitative central limit theorem in 1-Wasserstein distance. The proof uses a second Wiener chaos representation and the Malliavin-Stein method (in particular, the Fourth Moment Theorem), reducing normal approximation to (i) variance asymptotics and (ii) an explicit control of the fourth cumulant via trace estimates for an associated integral operator. For convenience, we provide self-contained kernel estimates (Dirichlet-type bounds, convolution inequalities, and a weighted Schur test) used in the argument.

Quantitative central limit theorem for an integrated periodogram via the fourth moment theorem

Abstract

We revisit the central limit theorem for integrated periodograms, equivalently for Toeplitz quadratic forms of stationary Gaussian sequences. Under a regular-variation assumption allowing long-memory singularities and slowly varying corrections, we prove a quantitative central limit theorem in 1-Wasserstein distance. The proof uses a second Wiener chaos representation and the Malliavin-Stein method (in particular, the Fourth Moment Theorem), reducing normal approximation to (i) variance asymptotics and (ii) an explicit control of the fourth cumulant via trace estimates for an associated integral operator. For convenience, we provide self-contained kernel estimates (Dirichlet-type bounds, convolution inequalities, and a weighted Schur test) used in the argument.

Paper Structure

This paper contains 7 sections, 11 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Model setting and main result
  3. Application to the Whittle estimator
  4. Proof of the main result
  5. Variance control
  6. Contraction / fourth cumulant
  7. Technical lemmas

Key Result

Theorem 2.1

Let $f,g\in L^{1}(-\pi,\pi)$ such that where $L_f$ and $L_g$ are slowly varying functions at $0$. Assume that Then we get Assume moreover that there exists $C>0$ such that, for all $\lambda,\omega$ satisfying $|\lambda-\omega|<\tfrac{|\lambda|}{2}$, Then, for every $\eta>0$, there exists a constant $C_\eta<\infty$ such that, for all $n$ large enough, $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (24)

  • Theorem 2.1
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 4.1: Second-chaos representation
  • proof
  • Proposition 4.2
  • proof
  • Proposition 4.3: Contraction control
  • proof
  • ...and 14 more