On the rate of convergence to the normal distribution of the number of false solutions of a system of nonlinear random Boolean equations
Authors:
V. I. Masol and S. Ya. Slobodyan
Translated by:
S. Kvasko
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom 76 (2007).
Journal:
Theor. Probability and Math. Statist. 76 (2008), 117129
MSC (2000):
Primary 60C05, 15A52, 15A03
Published electronically:
July 16, 2008
MathSciNet review:
2368744
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: We prove a theorem on the limit normal distribution (as ) of the number of false solutions of a system of nonlinear equations with independent random coefficients belonging to the field GF(2). We assume that every equation contains at least one coefficient for which the probability that it attains the value 1 is close to ; the number of equations and the number of unknowns are such that as ; the system has a solution containing units and as .
 1.
V.
G. Mikhaĭlov, Limit theorems for the number of nonzero
solutions of a system of random equations over the field
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
(2000b:11096), http://dx.doi.org/10.1137/S0040585X97977082
 2.
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
(94h:60089), http://dx.doi.org/10.1515/rose.1993.1.2.171
 3.
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
(2000f:60040), http://dx.doi.org/10.1137/S0040585X97976672
 1.
 V. G. Mikhaĭlov, Limit theorems for the number of nonzero solutions of a system of random equations over the field GF(2), Teor. Veroyatnost. i Primenen. 43 (1998), no. 3, 598606; English transl. in Theory Probab. Appl. 43 (1999), no. 3, 480487. MR 1681052 (2000b:11096)
 2.
 V. I. Masol, Moments of the number of solutions of a system of random Boolean equations, Random Oper. Stochastic Equations 1 (1993), no. 2, 171179. MR 1254185 (94h:60089)
 3.
 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, 4156; English transl. in Theory Probab. Appl. 43 (1999), no. 1, 7588. MR 1669972 (2000f:60040)
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Additional Information
V. I. Masol
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
vimasol@ukr.net
S. Ya. Slobodyan
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
sv_yaros@rambler.ru
DOI:
http://dx.doi.org/10.1090/S0094900008007369
PII:
S 00949000(08)007369
Keywords:
Nonlinear random Boolean equations,
the limit normal distribution,
the number of false solutions
Received by editor(s):
March 22, 2006
Published electronically:
July 16, 2008
Article copyright:
© Copyright 2008
American Mathematical Society
