Limiting behavior of largest entry of random tensor constructed by high-dimensional data

Abstract

Let ${X}_{k}=(x_{k1}, \cdots, x_{kp})', k=1,\cdots,n$, be a random sample of size $n$ coming from a $p$-dimensional population. For a fixed integer $m\geq 2$, consider a hypercubic random tensor $\mathbf{{T}}$ of $m$-th order and rank $n$ with \begin{eqnarray*} \mathbf{{T}}= \sum_{k=1}^{n}\underbrace{{X}_{k}\otimes\cdots\otimes {X}_{k}}_{m~multiple}=\Big(\sum_{k=1}^{n} x_{ki_{1}}x_{ki_{2}}\cdots x_{ki_{m}}\Big)_{1\leq i_{1},\cdots, i_{m}\leq p}. \end{eqnarray*} Let $W_n$ be the largest off-diagonal entry of $\mathbf{{T}}$. We derive the asymptotic distribution of $W_n$ under a suitable normalization for two cases. They are the ultra-high dimension case with $p\to\infty$ and $\log p=o(n^{\beta})$ and the high-dimension case with $p\to \infty$ and $p=O(n^{\alpha})$ where $\alpha,\beta>0$. The normalizing constant of $W_n$ depends on $m$ and the limiting distribution of $W_n$ is a Gumbel-type distribution involved with parameter $m$.