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

Arup Bose, Arnab Sen
2007 Electronic Communications in Probability  
Consider the matrix ∆ 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.
doi:10.1214/ecp.v12-1266 fatcat:wxt3dmopqnba3kb2vnznu57iwm