Note_1

In addition, since we showed in part (i) that this chain is an irreducible chain, hence each state communicate with each other state, hence all states must be of the same type since all states are in the same communication class (Theorem 5.3.2). Then if one state is aperiodic, then the all states that communicate with it must also be aperiod (to be of the same type). Hence in an irreduible chain, if one state is aperiodic, then all states are aperiodic as well.

Note_2

I allready had the code for the simulation written, just needed to feed the new data for this problem.