CT4 chapter 3 period of a state

Dear team ,

Can you please explain period and aperiod ?? from study material its confusing

regards

Suresh sharma

 

To work out the period of a state, you work out the possible return times to that state, take the highest common factor of them, and that's the period. The period must be greater than or equal to 2, otherwise the state is aperiodic.

So, if we can return to State 1 at times 2,4,6,8..., then the period of State 1 is 2 (the highest common factor of 2,4,6,8...).
If we can return to State 2 at times 4,6,8..., then the period of State 2 is 2 (the highest common factor of 4,6,8...).
If we can return to State 3 at times 2,3,4,5,6..., then since the highest common factor of 2,3,4,5,6... is 1, we say State 3 is aperiodic.
If we can return to State 4 at times 1,2,3,4,5,6...(eg there is a loop on State 4), then State 4 is also aperiodic.

If a Markov chain is irreducible, all its states will have the same period.
If a Markov chain is not irreducible, the states may have different periods.

 

thank you got it