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

TL;DR

This paper establishes a quantitative central limit theorem for integrated periodograms of stationary Gaussian sequences, incorporating long-memory effects, using advanced probabilistic techniques.

Contribution

It provides a new quantitative CLT result for Toeplitz quadratic forms with long-memory, employing the Fourth Moment Theorem and explicit kernel estimates.

Findings

Proves a CLT in 1-Wasserstein distance under long-memory assumptions.

Reduces normal approximation to variance asymptotics and cumulant control.

Provides explicit kernel estimates for the proof.

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.