Fixed point theorem for an infinite Toeplitz matrix

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.Comment: Revised, according to reviewer's comments. Conditionally accepted in The Bulletin of the Australian Mathematical Societ