On the Theory of Markoff Chains

Although there exists voluminous literature on the theory of probability of independent events, and powerful techniques have been developed for the analysis of most of the interesting problems in this field, the theory of probability of dependent events has been rather neglected. The first detailed investigations in this subject were published by A. Markoff 1. S. Bernstein 2 has extended the fundamental limit theorems to chains of dependent events. The most extensive exposition of this field has been made by M. Frechet 3. In the present paper we shall develop methods of averaging functions over chains of dependent variables and find the probability distribution of these functions. It will be shown that for certain types of chains these averages and distribution functions can be expressed in terms of the characteristic values and vectors of a certain operator equation. Many of the methods discussed here have been applied to problems in statistical mechanics 4,5,6,7,8. The most important application has been made by L. Onsager 8 who proved rigorously (on the basis of a simplified model) that Boltzmann's energy distribution in a solid with cooperative elements leads to a phase transition. The first explicit application of linear operator theory (through matrices and integral equations) to probability chains has apparently been made by Hostinsky 9.