On spectral distribution of sample covariance matrices from large dimensional and large k-fold tensor products
| Autor: | Benoît Collins, Jianfeng Yao, Wangjun Yuan |
|---|---|
| Rok vydání: | 2022 |
| Předmět: |
60B20
Statistics and Probability quantum information theory Probability (math.PR) FOS: Physical sciences Mathematics - Statistics Theory Mathematical Physics (math-ph) Statistics Theory (math.ST) Mathematics::Spectral Theory 15B52 eigenvalue distribution FOS: Mathematics Mathematics - Combinatorics Combinatorics (math.CO) Statistics Probability and Uncertainty Marčenko-Pastur law Mathematics - Probability Mathematical Physics large k-fold tensors |
| Zdroj: | Electronic Journal of Probability. 27 |
| ISSN: | 1083-6489 |
| DOI: | 10.1214/22-ejp825 |
| Popis: | We study the eigenvalue distributions for sums of independent rank-one $k$-fold tensor products of large $n$-dimensional vectors. Previous results in the literature assume that $k=o(n)$ and show that the eigenvalue distributions converge to the celebrated Mar\v{c}enko-Pastur law under appropriate moment conditions on the base vectors. In this paper, motivated by quantum information theory, we study the regime where $k$ grows faster, namely $k=O(n)$. We show that the moment sequences of the eigenvalue distributions have a limit, which is different from the Mar\v{c}enko-Pastur law. As a byproduct, we show that the Mar\v{c}enko-Pastur law limit holds if and only if $k=o(n)$ for this tensor model. The approach is based on the method of moments. Comment: 21 pages, 6 figures |
| Databáze: | OpenAIRE |
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