Full randomness in the higher difference structure of two-state Markov chains

Abstract

The paper studies the higher-order absolute differences taken from progressive terms of time-homogeneous binary Markov chains. Two theorems presented are the limiting theorems for these differences, when their order k converges to infinity. Theorems 1 and 2 assert that there exist some infinite subsets E of natural series such that kth order differences of every such chain converge to the equi-distributed random binary process as k growth to infinity remaining on E. The chains are classified into two types, and E depends only on the type of the given chain. Two kinds of discrete capacities for subsets of natural series are defined, and in their terms such sets E are described.