On asymptotic properties of the rank of a special random adjacency matrix

Arup Bose, Arnab Sen

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Consider the matrix Δn = ((I(Xi + Xj > 0)))i, j=1,2., n where (Xi) are i.i.d. and their distribution is continuous and symmetric around 0. We show that the rank rn of this matrix is equal in distribution to 2 Σn−1i=1 I(ξi = 1, ξi+1 = 0) + I(ξn = 1) (where ξii.i.d. ∼ Ber(1, 1/2). As a consequence √n(rn/n−1/2) is asymptotically normal with mean zero and variance 1/4. We also show that n−1rn converges to 1/2 almost surely.

Original languageEnglish (US)
Pages (from-to)200-205
Number of pages6
JournalElectronic Communications in Probability
Volume12
DOIs
StatePublished - Jan 1 2007

Keywords

  • 1-dependent sequence
  • Almost sure convergence
  • Almost sure representation
  • Convergence in distribution
  • Large dimensional random matrix
  • Rank

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