Marchenko-Pastur laws for Daniell smoothed periodograms

Abstract

Given a sample $X_0,...,X_{n-1}$ from a $d$-dimensional stationary time series $(X_t)_{t \in \mathbb{Z}}$, the most commonly used estimator for the spectral density matrix $F(\theta)$ at a given frequency $\theta \in [0,2\pi)$ is the Daniell smoothed periodogram $$S(\theta) = \frac{1}{2m+1} \sum\limits_{j=-m}^m I\Big( \theta + \frac{2\pi j}{n} \Big) \ ,$$ which is an average over $2m+1$ many periodograms at slightly perturbed frequencies. We prove that the Marchenko-Pastur law holds for the eigenvalues of $S(\theta)$ uniformly in $\theta \in [0,2\pi)$, when $d$ and $m$ grow with $n$ such that $\frac{d}{m} \rightarrow c>0$ and $d\asymp n^{\alpha}$ for some $\alpha \in (0,1)$. This demonstrates that high-dimensional effects can cause $S(\theta)$ to become inconsistent, even when the dimension $d$ is much smaller than the sample size $n$. Notably, we do not assume independence of the $d$ components of the time series. The Marchenko-Pastur law thus holds for Daniell smoothed periodograms, even when it does not necessarily hold for sample auto-covariance matrices of the same processes.