Estimates for the probability that a system of random equations is solvable in a given set of vectors over the field

Authors:
V. I. Masol and L. O. Romashova

Translated by:
S. Kvasko

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **87** (2012).

Journal:
Theor. Probability and Math. Statist. **87** (2013), 135-152

DOI:
https://doi.org/10.1090/S0094-9000-2014-00909-6

Published electronically:
March 21, 2014

MathSciNet review:
3241451

Full-text PDF

Abstract | References | Additional Information

Abstract: Let be the probability that a second order system of nonlinear random equations over the field has a solution in a given set of vectors, where is the number of unknowns in the system. A necessary and sufficient condition is found for as . Some rates of convergence to zero are found and some applications are described.

**1.**V. A. Kopyttsev and V. G. Mikhaĭlov,*Poisson-type theorems for the number of special solutions of a random linear inclusion*, Diskret. Mat.**22**(2010), no. 2, 3–21 (Russian, with Russian summary); English transl., Discrete Math. Appl.**20**(2010), no. 2, 191–211. MR**2730124**, https://doi.org/10.1515/DMA.2010.011**2.**V. I. Masol and L. A. Romaschova,*Uniqueness conditions for the solution of an inhomogeneous system of nonlinear random equations over the field 𝐺𝐹(3)*, Kibernet. Sistem. Anal.**46**(2010), no. 2, 23–36 (Russian, with Russian summary); English transl., Cybernet. Systems Anal.**46**(2010), no. 2, 185–199. MR**2921497**, https://doi.org/10.1007/s10559-010-9197-y**3.**K. A. Rybnikov,*\cyr Vvedenie v kombinatornyĭ analiz*, 2nd ed., Moskov. Gos. Univ., Moscow, 1985 (Russian). MR**812275****4.**William Feller,*An introduction to probability theory and its applications. Vol. I*, Third edition, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0228020****5.**Albert N. Shiryaev,*Problems in probability*, Problem Books in Mathematics, Springer, New York, 2012. Translated by Andrew Lyasoff. MR**2961901**

Additional Information

**V. I. Masol**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4E, Kiev 03127, Ukraine

**L. O. Romashova**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4E, Kiev 03127, Ukraine

Email:
deezee@ukr.net

DOI:
https://doi.org/10.1090/S0094-9000-2014-00909-6

Keywords:
System of nonlinear random equations,
probability that a system is solvable,
rate of convergence,
a field containing three elements

Received by editor(s):
July 4, 2011

Published electronically:
March 21, 2014

Article copyright:
© Copyright 2014
American Mathematical Society