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

Let Xk=(xk1,…,xkp)′,k=1,…,n, be a random sample of size n coming from a p-dimensional population. For a fixed integer m≥ 2 , consider a hypercubic random tensor T of mth order and rank n with T=∑k=1nXk⊗⋯⊗Xk⏟multiplicitym=(∑k=1nxki1xki2⋯xkim)1≤i1,…,im≤p.Let W_n be the largest off-diagonal entry of 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→ ∞ and log p= o(n^β) and the high-dimension case with p→ ∞ and p= O(n^α) where α, β> 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.