Regular Transition Probability Matrix

Definition

A TPM $\mathbf{P}$ is regular if:

1. For every pair $i,j$, there is a path $k_1,k_2,\dots ,k_r$ for which $P_{i{k}_1}{P}_{k_1{k}_2}\dots P_{k_{r}j}>0$
2. There is at least one state $i$ for which $P_{ii}>0$

Equivalently, ${\mathbf{P}}^k$ has all positive entries for some $k>0$.

Properties

1. If ${\mathbf{P}}^k$ has no zero elements for some $k$, then ${\mathbf{P}}^{k+n}$ will also have no zero elements for all $n\ge 1$
2. A regular chain is necessarily irreducible and aperiodic

Why Condition 2 Matters

Condition 2 ensures the chain is aperiodic.

Without it, a chain could be irreducible (condition 1 satisfied) but still periodic — meaning it cycles through states in a fixed pattern and never settles into a stationary distribution. A self-loop $P_{ii}>0$ forces the chain to be able to “stay” at a state, breaking any fixed cycle and guaranteeing period 1.

Together: irreducible + aperiodic = regular = guaranteed convergence to unique stationary distribution regardless of initial state.

Proof

See Regular chain is necessarily irreducible and aperiodic.

Related

• Transition Probability Matrix
• Irreducible
• Period (Stochastic)