A random walk Y_nn≧0 on a finite Markov chain X_nn≧0 with the state space   is considered:   is a family of random variables being independent of X_n and identically distributed in every sequence  . There are given criteria for the random walk Y_n to be not degenerated and for supremum   to be finite. The Fourier transforms of the limit functionals   are expressed in terms of the components of factorization of the matrix  . Some upper bounds for moments and distributions of such limit functionals are given.