Topics in the theory of Markoff chains

Abstract

Then pii(t) can be considered a transition probability of a Markoff chain: A system is supposed which can assume various numbered states, and pij(t) is the probability that the system is in the jth state at the end of a time interval of length t, if it was in the ith state at the beginning of the interval. The present paper will be divided into two parts. In the first, the regularity properties of P(t), and its asymptotic properties as t->O, t-> oo are studied. These problems have been solved in the finite-dimensional case by Doeblin(l). In the infinite-dimensional case new situations can arise, and the results are somewhat different. The method of approach is new, depending on two theorems (Theorems 2 and 3) concerning matrices whose elements are non-negative, and which have row sums less than or equal to 1. The method of approach can also be applied to the study of the asymptotic properties of the powers of a matrix of non-negative elements, with row sums 1. In the second part of the paper, the actual transitions connected with Markoff chains are investigated: That is, the properties of the function t(t), the number of the state which the given system assumes at time t, are investigated. The continuity properties of t(t) are analyzed, and related to the regularity properties of the pii(t).