This paper deals with limit distributions for sums $\eta _n $ which become independent when a certain path $x_n $, $n = 0,1,2, \cdots $, of a Markov chain is defined. The dependence between $\{ \eta _n \}$ and $\{ X_n \}$ is expressed more exactly by (1). Let $X_s $ be the path of a continuous Markov process. Furthermore, the study of the limit distributions of $\xi (t) = \int_0^t f ( X_s )ds$ at $t \to \infty $ can be reduced to the study of limit distributions of sums $\eta _n $. This reduction is illustrated for the case where $X_s $ is a one-dimensional diffusion process. The limit distribution for $\xi (t)$ coincides with distributions obtained in [12]. The sufficient conditions for convergence to each distribution are also given (Theorems 2 and 3).