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

Tiefeng Jiang, Junshan Xie

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish (US)
Pages (from-to)2380-2400
Number of pages21
JournalJournal of Theoretical Probability
Volume33
Issue number4
DOIs
StatePublished - Dec 1 2020

Bibliographical note

Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

  • Extreme-value distribution
  • High-dimensional data
  • Stein–Chen Poisson approximation
  • Tensor

Fingerprint

Dive into the research topics of 'Limiting Behavior of Largest Entry of Random Tensor Constructed by High-Dimensional Data'. Together they form a unique fingerprint.

Cite this