TY - JOUR
T1 - On asymptotic properties of the rank of a special random adjacency matrix
AU - Bose, Arup
AU - Sen, Arnab
PY - 2007/1/1
Y1 - 2007/1/1
N2 - 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 Σn−1i=1 I(ξi = 1, ξi+1 = 0) + I(ξn = 1) (where ξii.i.d. ∼ Ber(1, 1/2). As a consequence √n(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.
AB - 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 Σn−1i=1 I(ξi = 1, ξi+1 = 0) + I(ξn = 1) (where ξii.i.d. ∼ Ber(1, 1/2). As a consequence √n(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.
KW - 1-dependent sequence
KW - Almost sure convergence
KW - Almost sure representation
KW - Convergence in distribution
KW - Large dimensional random matrix
KW - Rank
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U2 - 10.1214/ECP.v12-1266
DO - 10.1214/ECP.v12-1266
M3 - Article
AN - SCOPUS:34249892159
SN - 1083-589X
VL - 12
SP - 200
EP - 205
JO - Electronic Communications in Probability
JF - Electronic Communications in Probability
ER -