Eigenvalue distribution of large sample covariance matrices of linear processes

Abstract

We derive the distribution of the eigenvalues of a large sample covariance matrix when the data is dependent in time. More precisely, the dependence for each variable $i=1,...,p$ is modelled as a linear process $(X_{i,t}{)}_{t=1,...,n}=({\sum }_{j=0}^{\infty }{c}_j{Z}_{i,t-j}{)}_{t=1,...,n}$, where $\{Z_{i,t}\}$ are assumed to be independent random variables with finite fourth moments. If the sample size $n$ and the number of variables $p=p_n$ both converge to infinity such that $y={\lim }_{n\to \infty }n/p_n>0$, then the empirical spectral distribution of $p^{-1}\X\X^T$ converges to a non\hyp{}random distribution which only depends on $y$ and the spectral density of $(X_{1,t}{)}_{t\in \mathbb{Z}}$. In particular, our results apply to (fractionally integrated) ARMA processes, which we illustrate by some examples.

12 pages

References in corpus (1)

• A Random Matrix Approach to VARMA Processes

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• Limit Theory for the largest eigenvalues of sample covariance matrices with heavy-tails
• Derivation of the Asymptotic Eigenvalue Distribution for Causal 2D-AR Models under Upscaling
• Spectral analysis of linear time series in moderately high dimensions