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 Wn be the largest off-diagonal entry of T. We derive the asymptotic distribution of Wn under a suitable normalization for two cases. They are the ultra-high-dimension case with p→ ∞ and log p= o(nβ) and the high-dimension case with p→ ∞ and p= O(nα) where α, β> 0. The normalizing constant of Wn depends on m and the limiting distribution of Wn is a Gumbel-type distribution involved with parameter m.
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- Extreme-value distribution
- High-dimensional data
- Stein–Chen Poisson approximation