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Teor. Veroyatnost. i Primenen., 1972, Volume 17, Issue 2, Pages 354–359 (Mi tvp2591)  

This article is cited in 2 scientific papers (total in 2 papers)

Short Communications

On the distribution of the linear rank of a random matrix

I. N. Kovalenko

Kiev

Abstract: 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\{-\frac1n\sum_{j=1}^nx_{ij}\},\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,…$$

Let $\alpha=1+\beta/\ln n$, $\beta$ being a bounded variable. Put
$$ \mu=e^{-\beta}\frac1n\sum_{j=1}^n\exp\{-\frac1n\sum_{i=1}^Nx_{ij}\}. $$
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$.

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English version:
Theory of Probability and its Applications, 1973, 17:2, 342–346

Bibliographic databases:

Received: 12.12.1969

Citation: I. N. Kovalenko, “On the distribution of the linear rank of a random matrix”, Teor. Veroyatnost. i Primenen., 17:2 (1972), 354–359; Theory Probab. Appl., 17:2 (1973), 342–346

Citation in format AMSBIB
\Bibitem{Kov72}
\by I.~N.~Kovalenko
\paper On the distribution of the linear rank of a~random matrix
\jour Teor. Veroyatnost. i Primenen.
\yr 1972
\vol 17
\issue 2
\pages 354--359
\mathnet{http://mi.mathnet.ru/tvp2591}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=297003}
\zmath{https://zbmath.org/?q=an:0253.60023}
\transl
\jour Theory Probab. Appl.
\yr 1973
\vol 17
\issue 2
\pages 342--346
\crossref{https://doi.org/10.1137/1117037}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Neuenschwander D., “Introduction – Background”, Probabilistic and Statistical Methodes in Cryptology: Introduction By Selected Topics, Lecture Notes in Computer Science, 3028, 2004, 1–155  mathscinet  isi
    2. V. I. Kruglov, V. G. Mikhailov, “O range sluchainoi dvoichnoi matritsy s zadannymi vesami nezavisimykh strok”, Matem. vopr. kriptogr., 10:4 (2019), 67–76  mathnet  crossref
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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