  For an infinite Toeplitz matrix $T$ with nonnegative real entries we find the
conditions, under which the equation $\boldsymbol{x}=T\boldsymbol{x}$, where
$\boldsymbol{x}$ is an infinite vector-column, has a nontrivial bounded
positive solution. The problem studied in this paper is associated with the
asymptotic behavior of convolution type recurrence relations, and can be
applied to different problems arising in the theory of stochastic processes and
applied problems from other areas.
