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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

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

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:

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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• http://mi.mathnet.ru/eng/tvp/v17/i2/p354

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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