In this paper, we mainly studied the limit properties for the
countable nonhomogeneous Markov chains. We established some limit properties for the
functions of the countable nonhomogeneous Markov chains with  variables under the
convergence in the  sense, which extended the similar conclusions for the
functions with two variables. At last, as a corollary, we given the similar result in
the homogeneous Markov stock market.