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

Abstract

Consider the matrix $\Delta_n = ((\ \mathrm{I}(X_i + X_j > 0)\ ))_{i,j = 1,2,...,n}$ 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 $2\sum_{i=1}^{n-1}\mathrm{I}(\xi_i =1,\xi_{i+1}=0)+\mathrm{I}(\xi_n=1)$ where $\xi_i \stackrel{i.i.d.}{\sim} \text{Ber} (1,1/2).$ As a consequence $\sqrt n(r_n/n-1/2)$ is asymptotically normal with mean zero and variance $1/4$. We also show that $n^{-1}r_n$ converges to $1/2$ almost surely.