Statistical and spectral properties of irreducible Markov chains

The spectral and high order statistical characteristics of irreducible Markov chains are studied. A Markov chain, which corresponds to a single-class state-space, is completely determined by its transition matrix. The paper explains analytically how the type of probability density function (PDF), describing the Markov chain, is determined by its transition matrix. Furthermore, it is shown that the correlation properties (power spectrum) and the high order spectra (HOS) of the Markov chain are also expressed by means of specific terms of the transition matrix.