The Invariance Principle for p-\ifmmode\expandafter\else.\expandafter\vecabove\fi. Chain

There are two parts in this paper. In the first part we construct the Markov chain in random environment (MCRE), the skew product Markov chain and p.\ifmmode\expandafter\else.\expandafter\vecabove\fi. chain from a random transition matrix and a two.dimensional probability distribution, and in the second part we prove that the invariance principle for p.\ifmmode\expandafter\else.\expandafter\vecabove\fi. chain, a more complex non.homogeneous Markov chain, is true under some reasonable conditions. This result is more powerful.