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


 
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1. Title Title of document On asymptotic properties of the rank of a special random adjacency matrix
 
2. Creator Author's name, affiliation, country Arup Bose; Indian Statistical Institute
 
2. Creator Author's name, affiliation, country Arnab Sen; University of California, Berkeley
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Large dimensional random matrix, rank, almost sure representation, $1$-dependent sequence, almost sure convergence, convergence in distribution.
 
3. Subject Subject classification Primary 60F99, Secondary 60F05, 60F15
 
4. Description 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.
 
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7. Date (YYYY-MM-DD) 2007-06-03
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1266
 
10. Identifier Digital Object Identifier 10.1214/ECP.v12-1266
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 12
 
12. Language English=en
 
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