Original article at: http://www.math.washington.edu/~ejpecp/ECP/viewarticle.php?id=1872

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

Arup Bose, Indian Statistical Institute
Arnab Sen, University of California, Berkeley

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∑i=1n-1 I(ξi =1,ξi+1=0)+I(ξn=1) where ξi are i.i.d. Ber(1,1/2).
   As a consequence n-1/2(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.

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Original article at: http://www.math.washington.edu/~ejpecp/ECP/viewarticle.php?id=1872