Limiting Behavior of Largest Entry of Random Tensor Constructed by High-Dimensional Data

Let $${X}_{k}=(x_{k1}, \ldots , x_{kp})', k=1,\ldots ,n$$ X k = ( x k 1 , … , x kp ) ′ , k = 1 , … , n , be a random sample of size n coming from a p-dimensional population. For a fixed integer $$m\ge 2$$ m ≥ 2 , consider a hypercubic random tensor $$\mathbf {{T}}$$ T of mth order and rank n with $$\begin{aligned} \mathbf {{T}}= \sum _{k=1}^{n}\underbrace{{X}_{k}\otimes \cdots \otimes {X}_{k}}_{\mathrm{multiplicity}\ m}=\Big (\sum _{k=1}^{n} x_{ki_{1}}x_{ki_{2}}\cdots x_{ki_{m}}\Big )_{1\le i_{1},\ldots , i_{m}\le p}. \end{aligned}$$ T = ∑ k = 1 n X k ⊗ ⋯ ⊗ X k ⏟ multiplicity m = ( ∑ k = 1 n x k i 1 x k i 2 ⋯ x k i m ) 1 ≤ i 1 , … , i m ≤ p . Let $$W_n$$ W n be the largest off-diagonal entry of $$\mathbf {{T}}$$ T . We derive the asymptotic distribution of $$W_n$$ W n under a suitable normalization for two cases. They are the ultra-high-dimension case with $$p\rightarrow \infty $$ p → ∞ and $$\log p=o(n^{\beta })$$ log p = o ( n β ) and the high-dimension case with $$p\rightarrow \infty $$ p → ∞ and $$p=O(n^{\alpha })$$ p = O ( n α ) where $$\alpha ,\beta >0$$ α , β > 0 . The normalizing constant of $$W_n$$ W n depends on m and the limiting distribution of $$W_n$$ W n is a Gumbel-type distribution involved with parameter m.