Investigation of the solutions of linear systems of difference equations with random coefficients (Q1280279)

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.