Limiting moments of sample covariance matrices from large fold tensor vectors

Source abstract

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 $p \in \mathbb N_+$, under suitable moment conditions, we obtain an asymptotic formula for $p$th moment as $n, k, m$ tend to infinity with $m/n^k \to c \in (0,\infty)$. 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 $p$th moment of normalized matrices. In this regime, we also provide a lower bound for the growth rate of the largest eigenvalue.