Irreducibility¶

Definition¶

For a homogeneous Markov chain, state $j$ is accessible from state $i$ if $(\boldsymbol{P}^n)_{ij}>0$ for some integer $n\ge0$$0$. Two states communicate if each is accessible from the other. The chain is irreducible when every pair of states communicates.

Interpretation¶

Accessibility uses the multi-step probabilities of the Markov chain. A direct transition can have probability zero while a route through intermediate states has positive probability. Irreducibility says that all states form a single communicating class, that is, a maximal set of states that communicate with each other; it does not require every matrix entry to be positive.

Key Results¶

• A finite irreducible chain has a unique stationary distribution, with positive probability on every state.
• Irreducibility alone does not ensure convergence of the distribution at a fixed time: a chain can still be periodic.
• For a finite chain, draw an arrow $i\to j$ when $p_{ij}>0$$0$. Irreducibility is equivalent to a directed path between every ordered pair of states.

Examples¶

The two-state alternating chain $\boldsymbol{P}=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ is irreducible: either state reaches the other in one step. It has period 2. By contrast, $\boldsymbol{P}=\boldsymbol{I}$ with two states is reducible because neither state can reach the other.

Sources¶

• MIT Introduction to Probability, Lectures 24–25.
• Course placement: RUG Stochastic Models, 2026–2027. These foundation notes have no assigned lecture yet.

Prerequisites¶

• Markov Chain

Related Concepts¶

• Stationary Distribution
• Aperiodicity