Are the states (or set of states) absorbing, transient or recurrent?

[CTK]

Summary: Determine the absorbing states & communication classes of the given matrix.

Hello everyone,

If we have a state space of S = {1,2,3,4} and the following matrix:

\begin{bmatrix}
0 & 1 & 0 & 0\\
0 & 0 & 1/3 & 2/3\\
1 & 0 & 0 & 0\\
0 & 1/2 & 1/2 & 0\\
\end{bmatrix}

Now, given the above, I don't think there are any absorbing states or sets of states, is that correct?
And since the above is a finite and irreducible closed set of states, then all of the states are recurrent and there are no transient states, right?
Also, since all the states communicate with each other, then the communication class is simply all of the state space, right?
Finally, not directly related to the above matrix, but just in general, if we don't have a limit law, then does that imply that we don't have a stationary distribution?

Please correct me if I am wrong, thanks.

 

[PeroK]

That looks right. Did you draw a map of the states' interactions?

 

[CTK]

That looks right. Did you draw a map of the states' interactions?

Yes, I did. But I am already struggling to type a matrix, let alone a graph ahahaha.
Just a quick follow up on the above matrix, what would the stationary distribution be for the above matrix in this case?
Also, what would the period of each state be? Because I got a period of 1 for every single state, but that's not possible because we just said above that it is periodic so it means the periodicity should be greater than 1, right?

Any suggestions would be appreciate it.

 

[PeroK]

For a stationary distribution you are looking for a "back to front" eigenvector of the matrix.

Nothing looks periodic.

 

[CTK]

For a stationary distribution you are looking for a "back to front" eigenvector of the matrix.

Nothing looks periodic.

I guess I have figured it out, thanks for your help, it is really appreciated. Have a good one.

 

[PeroK]

I guess I have figured it out, thanks for your help, it is really appreciated. Have a good one.

There should be one stationary state - associated with the eigenvalue $1$ of course. By "back to front" eigenvector, I meant eigenvector of the transpose of the transition matrix. I've no idea why Markov theory has matrices acting on the right of row vectors - rather than the usual way round by acting from the left on column vectors.

 

[CTK]

There should be one stationary state - associated with the eigenvalue $1$ of course. By "back to front" eigenvector, I meant eigenvector of the transpose of the transition matrix. I've no idea why Markov theory has matrices acting on the right of row vectors - rather than the usual way round by acting from the left on column vectors.

Yeah, in many cases, I am still not understanding the details but rather just implementing them to get an answer ahahaha. Thanks.