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 Teor. Veroyatnost. i Primenen., 1998, Volume 43, Issue 3, Pages 598–606 (Mi tvp1564)

Short Communications

Limit theorems for the number of nonzero solutions of a system of random equations over GF(2)

V. G. Mikhailov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The asymptotic behavior of a number of solutions of a system of random equations of a particular form over GF(2) is investigated. The left-hand sides of the equations of the system are products of independent equiprobable linear functions in $n$ variables for GF(2), whereas the right-hand sides are equal to zero. Under the natural restrictions on the way of changing the parameters of the scheme (the number of unknowns, the number of equations, and the number of multipliers in the left-hand side of each equation) it is shown that the distribution of the number of nonzero solutions converges to a Poisson distribution. Sufficient conditions are given for the number of nonzero solutions to be asymptotically normal. The proofs are based on the moment method.

Keywords: systems of random equations, number of solutions, Poisson distribution.

DOI: https://doi.org/10.4213/tvp1564

Full text: PDF file (470 kB)

English version:
Theory of Probability and its Applications, 1999, 43:3, 480–487

Bibliographic databases:

Citation: V. G. Mikhailov, “Limit theorems for the number of nonzero solutions of a system of random equations over GF(2)”, Teor. Veroyatnost. i Primenen., 43:3 (1998), 598–606; Theory Probab. Appl., 43:3 (1999), 480–487

Citation in format AMSBIB
\Bibitem{Mik98} \by V.~G.~Mikhailov \paper Limit theorems for the number of nonzero solutions of a~system of random equations over GF(2) \jour Teor. Veroyatnost. i Primenen. \yr 1998 \vol 43 \issue 3 \pages 598--606 \mathnet{http://mi.mathnet.ru/tvp1564} \crossref{https://doi.org/10.4213/tvp1564} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1681052} \zmath{https://zbmath.org/?q=an:0951.60011} \transl \jour Theory Probab. Appl. \yr 1999 \vol 43 \issue 3 \pages 480--487 \crossref{https://doi.org/10.1137/S0040585X97977082} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000085137400010} 

• http://mi.mathnet.ru/eng/tvp1564
• https://doi.org/10.4213/tvp1564
• http://mi.mathnet.ru/eng/tvp/v43/i3/p598

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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. V. G. Mikhailov, “Limit theorems for the number of nonzero solutions of a system of random equations over the field $\mathrm{GF}(2)$”, Discrete Math. Appl., 10:2 (2000), 115–126
2. V. G. Mikhailov, “The Poisson limit theorem for the number of noncollinear solutions of a system of random equations of a special form”, Discrete Math. Appl., 11:4 (2001), 391–400
3. V. G. Mikhailov, “Limit theorems for the number of points of a given set covered by a random linear subspace”, Discrete Math. Appl., 13:2 (2003), 179–188
4. V. G. Mikhailov, “Limit theorems for the number of solutions of a system of random linear equations belonging to a given set”, Discrete Math. Appl., 17:1 (2007), 13–22
5. V. A. Kopyttsev, V. G. Mikhailov, “Poisson-type limit theorems for the generalised linear inclusion”, Discrete Math. Appl., 22:4 (2012), 477–491
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