Limiting behavior of largest entry of random tensor constructed by high-dimensional data
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Let X_k=(x_k1, ..., x_kp)', 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 m-th order and rank n with T= ∑_k=1^nX_k⊗...⊗X_k_m multiple=(∑_k=1^n x_ki_1x_ki_2... x_ki_m)_1≤ i_1,..., i_m≤ 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.
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