What is strictly periodic cycle

Question

In AI book by Norvig and Russell ergodic Markov Chains are defined as follows:

1. Every state is reachable from every other.
2. There are no strictly periodic cycles in it.

Can someone explain what is strictly periodic cycle or give any reference?

Answer 1

After some reading on wikipedia I believe what they mean is that for any state $i$ and time $0$, the times $t$ where you can reach $i$ again is not limited to a multiple of any integer greater than $1$. For instance, a random walk is periodic with period $2$ (you cannot come back to where you started after an odd number of moves; it has to be even).

In contrast, say we have a normal 1-D random walk (infinite or finite), but with the single change that if the walker is on square $100$, he can also choose to stand still in addition to moving left or right. Then it's not periodic, since there's a positive probability that he'll be back at the origin at the odd time $201$, even though up until that point it has seemed periodic.

This is more an account of the wikipedia definition than it addresses your concern regarding the word strictly, but I hope it helps you some part along the way.