Limit distribution of the number of solutions of a system of stochastic linear homogeneous equations with a uniform coefficient matrix over a finite local ring
type: Item
instance of: scholarly article
title: Limit distribution of the number of solutions of a system of stochastic linear homogeneous equations with a uniform coefficient matrix over a finite local ring
zbMATH Open document ID: 0924.15016
DOI: 10.1007/BF02666988
published in: Cybernetics and Systems Analysis
publication date: 1999-06-10T00:00:00Z
review text: Let \(R\) be a finite local ring with radical \(I(R)\). Consider the system of linear equations (SLE): \[ (1)\quad Ax^\downarrow=0^\downarrow,\qquad (2)\quad By^\downarrow=0^\downarrow, \] where \(A\) and \(B\) are \((n+s)\times n\) matrices over \(R\) and \(s\) is an integer constant. Assume that the elements of each matrix \(A\) and \(B\) are jointly independent random variables uniformly distributed on the set \(R\) and \(I(R)\), respectively. Denote by \(\nu_{n,s}\) the number of solutions of SLE(1) and by \(\mu_{n,s}\) the number of solutions of SLE(2). Let \(s_0=-s\vee 0\), \(\pi_{n,s}(k)=P(\nu_{n,s}=q^k)\), \(\rho_{n,s}(k)= P(\mu_{n,s}(k)=q^k)\), \(k=0,1,\dots\). The exact expressions and the limit expressions (as \(n\to\infty)\) for the probabilities \(\pi_{n,s}(k)\), \(k=0,1, \dots\) and the limit (as \(n\to\infty)\) for \(\pi_{n,s}(-sN)\), where \(s\leq 0\) are obtained. The author proves a theorem which generalizes previous results.
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 60B12
zbMATH DE Number: 1283637
zbMATH Keywords: limit distribution
zbMATH Keywords: number of solutions
zbMATH Keywords: system of stochastic linear homogeneous equations
zbMATH Keywords: uniform coefficient matrix
zbMATH Keywords: finite local ring
author: A. N. Alekseychuk
reviewed by: Rodica Covaci
MaRDI profile type: Publication
cites work: Q5622122
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cites work: Matrices over a Finite Field
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description: scientific article; zbMATH DE number 1283637