  Let S be a denumerable state space and let P be a transition probability
matrix on S. If a denumerable set M of nonnegative matrices is such that the
sum of the matrices is equal to P, then we call M a partition of P. Let K
denote the set of probability vectors on S. To every partition M of P we can
associate a transition probability function on K defined in such a way that if
p in K and m in M are such that ||pm|| > 0, then, with probability ||pm|| the
vector p is transferred to the vector pm/||pm||. Here ||.|| denotes the
l_1-norm. In this paper we investigate convergence in distribution for Markov
chains generated by transition probability functions induced by partitions of
transition probability matrices. An important application of the convergence
results obtained is to filtering processes of partially observed Markov chains.
