The local limit theorem for Markov chains and regularity conditions

The following stochastic process $x_t$ is considered. Given a set of segments $\{\Delta_i=[a_i,b_i],\ i=\overline{1,s}\}$, a point $x_t$ is moving uniformly along $\Delta_i$, having reached $b_i$ it jumps to $a_j$ with a probability $p_{ij}$ and then it goes on moving uniformly. In the present paper the necessary and sufficient conditions of regularity of $x_t$ are obtained. At the same time the connexion between these conditions and those of the local limit theorem for finite Markov chains is established.