given this transition matrix of markov chain
\begin{bmatrix} \dfrac{1}{2} & \dfrac{1}{4} & \dfrac{1}{4}\\ 0 & \dfrac{1}{2} & \dfrac{1}{2} \\ 1 & 0 & 0 \end{bmatrix}
which represents transition matrix of states $a,b,c$.
$a$ has probability of $\dfrac{1}{2}$ to itself $\dfrac{1}{4}$ to $b$ $\dfrac{1}{4}$ to $c$.
b has probability $\dfrac{1}{2}$ to itself and $\dfrac{1}{2}$ to $c$
c has probability $1$ to $a$.
- why is state $c$ aperiodic?
I know that it is irreducible and state a is aperiodic because it has self loop so all states are aperiodic. but i can't see why states that don't have self loops are aperiodic.
if one can explain what exactly aperdicity and why state c is aperiodic from the definition of aperiodicity itself.