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On asymptotic properties of the rank of a special random adjacency matrix
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Arup Bose, Indian Statistical Institute
Arnab Sen, University of California, Berkeley
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Abstract
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∑i=1n-1 I(ξi =1,ξi+1=0)+I(ξn=1) where ξi
are i.i.d. Ber(1,1/2).
As a consequence n-1/2(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.
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