On the distribution of the linear rank of a random matrix

I. N. Kovalenko

Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 2, pp. 354-359

Résumé

Let $A=\|a_{ij}\|$ be a $N\times n$ random matrix, $a_{ij}$ being independent one-zero variables, $\mathbf P\{a_{ij}=1\}=\frac{\ln n+x_{ij}}{n}$, where $|x_{ij}|\le T$ for all possible $i$, $j$. Denote by $\xi$ the number of non-zero rows of the matrix $A$ and by $\eta$ the number of its non-zero columns and set $\zeta=\min\{\xi,\eta\}$. The purpose of this note is to investigate the limiting behaviour of $\zeta$'s distribution as $n\to\infty$. Put $$ \lambda=\frac1n\sum_{i=1}^N\exp\biggl\{-\frac1n\sum_{j=1}^nx_{ij}\biggr\},\quad\alpha=N/n. $$ Theorem 2 states that condition $n^\alpha(1-\alpha)\to\infty$ implies that $$ \mathbf P\{\zeta=\xi\}\to1,\quad\mathbf Р\{\zeta=N-k\}-e^{-\lambda}\frac{\lambda^k}{k!}\to0,\quad k=0,1,\dots $$ Let $\alpha=1+\beta/\ln n$, $\beta$ being a bounded variable. Put $$ \mu=e^{-\beta}\frac1n\sum_{j=1}^n\exp\biggl\{-\frac1n\sum_{i=1}^Nx_{ij}\biggr\}. $$ Then the distribution of the random variable $\zeta$ asymptotically coincides with that of the $\min\{N-U,n-V\}$, where $U$, $V$ are independent Poisson random variables with parameters $\lambda$, $\mu$.