Extension of the notion of irregularity and a class of limit theorems for nonhomogeneous markov chains

The purpose of this paper is to give a class of limit theorems for nonhomogeneous markov chains which contain an extension of the notion of the irregularity. The primary idea of the notion of irregularity is probability theory is that in Bernoulli sequences taking values 0 and 1 with equal possibility, the occurrence of 0 and 1 is irregular completely, and it is impossible to make the relative frequency of '1' greater or less than one of '0' by selection.