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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.14618 |
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Table of Contents:
- 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 $F(θ)$ at a given frequency $θ\in [0,2π)$ is the Daniell smoothed periodogram $$S(θ) = \frac{1}{2m+1} \sum\limits_{j=-m}^m I\Big( θ+ \frac{2πj}{n} \Big) \ ,$$ 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 $S(θ)$ uniformly in $θ\in [0,2π)$, when $d$ and $m$ grow with $n$ such that $\frac{d}{m} \rightarrow c>0$ and $d\asymp n^α$ for some $α\in (0,1)$. This demonstrates that high-dimensional effects can cause $S(θ)$ 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.