Aperiodicity¶

Definition¶

For a state $i$ of a homogeneous Markov chain with at least one positive-probability return, its period is

d(i)=\gcd\{n\ge1\colon(\boldsymbol{P}^n)_{ii}>0\}.

The state is aperiodic when $d(i)=1$ $1$ . An irreducible chain has the same period at every state, so it is enough to check one state. A chain is aperiodic when all its states have period 1.

Interpretation¶

The powers of the transition matrix identify possible return times. A period greater than one forces returns onto a repeating time grid. Irreducibility allows states to reach one another, which is why they share a period. Aperiodicity removes this cyclic obstruction; it does not require a positive one-step self-transition at every state.

Key Results¶

For a finite irreducible aperiodic chain, $\boldsymbol{\alpha}\boldsymbol{P}^n\to\boldsymbol{\pi}$ for every initial probability row vector $\boldsymbol{\alpha}$ , where $\boldsymbol{\pi}$ is its unique stationary distribution.
In an irreducible chain, a positive self-transition at one state implies period 1 at all states. This is a sufficient condition, not a necessary one.
Aperiodicity is not needed merely to solve the stationary equations, nor for uniqueness in a finite irreducible chain.

Examples¶

For $\boldsymbol{P}=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ $(0110)$ , returns occur only at times $2,4,6,\ldots$ , so the period is 2. Replacing the matrix by $\begin{pmatrix}1/2&1/2\\1/2&1/2\end{pmatrix}$ permits a return after one step and makes the chain aperiodic.

Sources¶

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

Prerequisites¶

Stationary Distribution

Aliases: period of a state Symbols: alpha, pi