On spectrum of sample covariance matrices from large tensor vectors

Wangjun Yuan

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arXiv · 2306.05834

On spectrum of sample covariance matrices from large tensor vectors

Wangjun Yuan

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Abstract

In this paper, we investigate the limiting empirical spectral distribution (LSD) of sums of independent rank-one $k$-fold tensor products of $n$-dimensional vectors as $k,n \to \infty$. Assuming that the base vectors are complex random variables with unit modular, we show that the LSD is the Mar\v{c}enko-Pastur law. Comparing with the existing results, our limiting setting allows $k$ to grow much faster than $n$. Consequently, we obtain the necessary and sufficient conditions for Mar\v{c}enko-Pastur law to serve as the LSD of our matrix model. Our approach is based on the moment method.

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Wangjun Yuan. 2023-06-09. On spectrum of sample covariance matrices from large tensor vectors. https://arxiv.org/abs/2306.05834

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