Limiting moments of sample covariance matrices from large fold tensor vectors
Abstract
We study the moments of sample covariance matrices Mn,k,m formed from independent k-fold tensor products of n-dimensional random vectors. The base vectors have independent and identically distributed centered complex entries with unit variance and finite moments of all orders. For every fixed p ∈ ℕ+, under suitable moment conditions, we obtain an asymptotic formula for pth moment as $n, k, m$ tend to infinity with m/nk → c ∈ (0,∞). There is no restriction on the growth rate of k relative to n. We also prove a variance bound and obtain the corresponding almost sure asymptotic formula. The results recover known limiting moments in several regimes. When k/n tends to infinity and the modulus of base vector entry is not constant, we establish the asymptotic growth rate for the pth moment of normalized matrices. In this regime, we also provide a lower bound for the growth rate of the largest eigenvalue.
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BibTeX
@misc{yuan2026limiting,
title = {{Limiting moments of sample covariance matrices from large fold tensor vectors}},
author = {Wangjun Yuan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33687},
eprint = {2610.33687},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33687},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Limiting moments of sample covariance matrices from large fold tensor vectors
AU - Wangjun Yuan
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33687
UR - https://arxiv.org/abs/2610.33687
AB - We study the moments of sample covariance matrices M_n,k,m formed from independent k-fold tensor products of n-dimensional random vectors. The base vectors have independent and identically distributed centered complex entries with unit variance and finite moments of all orders. For every fixed p ∈ ℕ_+, under suitable moment conditions, we obtain an asymptotic formula for pth moment as $n, k, m$ tend to infinity with m/n^k → c ∈ (0,∞). There is no restriction on the growth rate of k relative to n. We also prove a variance bound and obtain the corresponding almost sure asymptotic formula. The results recover known limiting moments in several regimes. When k/n tends to infinity and the modulus of base vector entry is not constant, we establish the asymptotic growth rate for the pth moment of normalized matrices. In this regime, we also provide a lower bound for the growth rate of the largest eigenvalue.
ER -