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

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 $mgeq 2$, 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}otimescdotsotimes {X}_{k}}_{m~multiple}=Big(sum_{k=1}^{n} x_{ki_{1}}x_{ki_{2}}cdots x_{ki_{m}}Big)_{1leq 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 $ptoinfty$ and $log p=o(n^{beta})$ and the high-dimension case with $pto infty$ 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 $m$.