# Markov chain, non-decomposable

A Markov chain whose transition probabilities $P_{ij}(t)$ have the following property: For any states $i$ and $j$ there is a time $t_{ij}$ such that $p_{ij}(t_{ij}) > 0$. The non-decomposability of a Markov chain is equivalent to non-decomposability of its matrix of transition probabilities $P = \left( {p_{ij}} \right)$ for a discrete-time Markov chain, and of its matrix of transition probability densities $Q = \left( {p'_{ij}(0)} \right)$ for a continuous-time Markov chain. The state space of a non-decomposable Markov chain consists of one class of communicating states (cf. Markov chain).