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

Abstract

Consider the matrix Δ n =(( I(X i +X j >0) )) i,j=1,2,...,n where {X i } are i.i.d.\ and their distribution is continuous and symmetric around 0 . We show that the rank r n of this matrix is equal in distribution to 2∑ n−1 i=1 I(ξ i =1,ξ i+1 =0)+I(ξ n =1) where ξ i ∼ i.i.d. Ber(1,1/2). As a consequence n − √ (r n /n−1/2) is asymptotically normal with mean zero and variance 1/4 . We also show that n −1 r n converges to 1/2 almost surely.