This paper deals with limit distributions for sums $\eta_n$ which become independent when a certain path $x_n$, $n=0,1,2,\dots$, 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 $\zeta(t)=\int_0^t f(X_s)\,ds$ 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 $\zeta(t)$ coincides with distributions obtained in [12]. The sufficient conditions for convergence to each distribution are also given (Theorems 2 and 3).