Period (Stochastic)

Definition

The period of state $i$ is defined as:

$d(i)=\gcd \{n\ge 1:P_{ii}^{(n)}>0\}$

The GCD of all steps $n$ at which it is possible to return to state $i$.

• Periodic: period $d>1$ — returns are confined to multiples of $d$
• Aperiodic: period $d=1$ — no such rhythm, can return at irregular times

Example

Periodic State (d=2)

A swing at the highest point on the right side:

• If at highest point now (Step 0), must swing down→left→back→right
• Can only be at that state at Step 2, 4, 6, etc.
• Period $d=2$

Aperiodic State (d=1)

Checking phone for notifications:

• Can return at Step 1, 5, 12, or any time
• GCD of all possible return times is 1

Related

• Aperiodic vs Periodic State
• Ergodic State
• Regular TPM