Investigation of the solutions of linear systems of difference equations with random coefficients
type: Item
instance of: scholarly article
title: Investigation of the solutions of linear systems of difference equations with random coefficients
zbMATH Open document ID: 0912.39001
DOI: 10.1023/A:1004361128462
author: G. S. Khalil
author: K. G. Valeev
published in: Periodica Mathematica Hungarica
publication date: 1999-03-14T00:00:00Z
review text: The authors study a linear system of difference equations \[ X_{n+1} =Q(y_n)X_n, \quad n=0,1,\dots \tag{\(*\)} \] Here \((y_n)\) is an i.i.d. sequence of nonnegative random variables having density \(\varphi\) and \(Q(r)\) is an \(m\times m\)-matrix which depends on a real number \(r\geq 0\). Let \(A\) denote the matrix given by \[ A=\int^\infty_0 Q(y)\varphi (y) dy. \] Then it is shown e.g. that the null solution of \((*)\) is asymptotically stable in the mean if and only if all eigenvalues of A have moduli less than one. Similar results are obtained in the case when the matrix \(Q\) depends on \(y_n\) and \(\zeta_n\). Here \((\zeta_n)\) is a Markov chain (independent of \((y_n)\)) with a finite state space and stationary transition probabilities.
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 60H25
zbMATH DE Number: 1261185
zbMATH Keywords: system of linear difference equations
zbMATH Keywords: random coefficients
reviewed by: Klaus Schürger
MaRDI profile type: Publication
full work available at URL: a:1004361128462
OpenAlex ID: W1837247627
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description: scientific article; zbMATH DE number 1261185