Normal limiting distribution of the normalized number of extraneous solutions of a compatible system of nonlinear random equations over the field GF(2)
Authors:
V. I. Masol and S. Ya. Slobodyan
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 90 (2015), 139-151
MSC (2010):
Primary 60C05, 15A52, 60F99
DOI:
https://doi.org/10.1090/tpms/955
Published electronically:
August 10, 2015
MathSciNet review:
3242026
Full-text PDF Free Access
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Additional Information
Abstract: Conditions are presented under which the distribution of the properly normalized number of extraneous solutions of a system of compatible random equations over the field GF(2) tends to a standard normal distribution.
References
- V. I. Masol, Moments of the number of solutions of a system of random Boolean equations, Random Oper. Stochastic Equations 1 (1993), no. 2, 171–179. MR 1254185, DOI https://doi.org/10.1515/rose.1993.1.2.171
- Volodymyr Masol and Svitlana Slobodyan, On the asymptotic normality of the number of false solutions of a system of nonlinear random Boolean equations, Theory Stoch. Process. 13 (2007), no. 1-2, 144–151. MR 2343819
- V. I. Masol, A theorem on the limit distribution of the number of false solutions of a system of nonlinear random Boolean equations, Teor. Veroyatnost. i Primenen. 43 (1998), no. 1, 41–56 (Russian, with Russian summary); English transl., Theory Probab. Appl. 43 (1999), no. 1, 75–88. MR 1669972, DOI https://doi.org/10.1137/S0040585X97976672
- V. G. Mikhaĭlov, Limit theorems for the number of nonzero solutions of a system of random equations over the field ${\rm GF}(2)$, Teor. Veroyatnost. i Primenen. 43 (1998), no. 3, 598–606 (Russian, with Russian summary); English transl., Theory Probab. Appl. 43 (1999), no. 3, 480–487. MR 1681052, DOI https://doi.org/10.1137/S0040585X97977082
- K. A. Rybnikov, Vvedenie v kombinatornyĭ analiz, 2nd ed., Moskov. Gos. Univ., Moscow, 1985 (Russian). MR 812275
References
- V. I. Masol, Moments of the number of solutions of system of random Boolean equations, Random Oper. Stoch. Eq. 1 (1993), no. 2, 171–179. MR 1254185 (94h:60089)
- V. I. Masol and S. Y. Slobodyan, On the asymptotic normality of the number of false solutions of a system of nonlinear random Boolean equations, Theory Stoch. Process. 13(29) (2007), no. 1–2, 144–151. MR 2343819 (2009a:60005)
- V. I. Masol, A theorem on the limiting distribution for the number of false solutions of a system of nonlinear random Boolean equations, Teor. Veroyatnost. Primenen. 43 (1998), no. 1, 41–56; English transl. in Theory Probab. Appl. 43 (1999), no. 1, 75–88. MR 1669972 (2000f:60040)
- V. G. Mikhaĭlov, Limit theorems for the number of nonzero solutions of a system of random equations over GF(2), Teor. Veroyatnost. Primenen. 43 (1998), no. 3, 598–606; English transl. in Theory Probab. Appl. 43 (1999), no. 3, 480–487. MR 1681052 (2000b:11096)
- K. A. Rybnikov, Introduction to Combinatorial Analysis, 2nd edition, Moscow University Press, Moscow, 1985. (Russian) MR 812275 (87b:05002)
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Additional Information
V. I. Masol
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine
Email:
vimasol@ukr.net
S. Ya. Slobodyan
Affiliation:
Department of Statistics and Higher Mathematics, Faculty for Mathematics and Informatics, Vasyl Stefanyk Precarpathian National University, Shevchenko Street, 57, Ivano-Frankivsk, 76018, Ukraine
Email:
slobodian_s@ukr.net
Keywords:
Random equations,
field GF(2),
limiting normal distribution,
extraneous solutions
Received by editor(s):
March 18, 2013
Published electronically:
August 10, 2015
Article copyright:
© Copyright 2015
American Mathematical Society
