Orthogonal cycle transforms of stochastic matrices

Abstract

Abstract. In this paper we investigate new Fourier series with respect to orthonormal families of directed cycles y, which occur in the graph of a recurrent stochastic matrix P. Specifically, it is proved that P may be approximated in a suitable Hilbert space by the Fourier series y~. w• This approach provides a proof in terms of Hilbert space of the cycle decomposition ~rrnula for finite stochastic matrices P. PreliminariesLet S be at most a denumerable set and let P = (Pij, i, j ~ S) be an irreducible and positive-recurrent stochastic matrix. Then there exists a probability row-distribution ~ = (zri, i 6 S) such that Jri > 0, i ~ S, and ~TCiPij = ETrjpji, i E S.(1)