If we have an irreducible Markov Chain in which any state has a self loop, it becomes aperiodic. This fact made sense to me, but it got me wondering about two things:
Is it possible for an irreducible Markov Chain to exist which is aperiodic but does NOT have a self loop?
Is it possible to have a non irreducible Markov Chain which is aperiodic? (I'm leaning towards a yes on this one). In particular, a non irreducible Markov Chain without any self loops which is aperiodic.
It would be great if someone could intuitively or with examples explain the above queries.
TIA!
Both of these are possible.
First, suppose we have a Markov chain with three states $A$, $B$, $C$ and a transition from any state to any other state (no loops). From state $A$ we can return back to $A$ in $2$ steps ($A \to B \to A$) or in $3$ steps ($A \to B \to C \to A$) and these have GCD $1$, so state $A$ is aperiodic; the same argument applies to other states.
Now take two copies of this Markov chain: states $A, B, C$ with transitions between any two of them, and three more states $A', B', C'$ with transitions between any two of them. This is still aperiodic for all the same reasons, but because we can't get from $\{A,B,C\}$ to $\{A', B', C'\}$, it's not irreducible.