@article{ECP1266,
author = {Arup Bose and Arnab Sen},
title = {On asymptotic properties of the rank of a special random adjacency matrix},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {12},
year = {2007},
keywords = {Large dimensional random matrix, rank, almost sure representation, \$1\$-dependent sequence, almost sure convergence, convergence in distribution.},
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.},
pages = {no. 20, 200-205},
issn = {1083-589X},
doi = {10.1214/ECP.v12-1266},
url = {http://ecp.ejpecp.org/article/view/1266}}