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

• Published in: Journal of theoretical probability 2020-12, Vol.33 (4), p.2380-2400
• Main Authors: Jiang, Tiefeng, Xie, Junshan
• Format: Article
• Language: English
• Subjects: Constraining; Mathematical analysis; Mathematics; Mathematics and Statistics; Normalizing; Probability Theory and Stochastic Processes; Statistics; Tensors
• ISSN: 0894-9840; 1572-9230
• DOI: 10.1007/s10959-019-00958-1

Let 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 ≥ 2 , consider a hypercubic random tensor T of m th order and rank n with 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 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 .