I've got this theorem that we covered in class, but I didn't get a chance to write down the proof. I'm not quite sure how to prove it.
I'm not sure how to go about proving this on my own, and I can't seem to find the proof online.
I can understand the theorem intuitively. A transient state is only visited a finite number of times, so as when we look at the nth state as n goes to infinity, it makes sense that the transient state will have had its last visit, and will not be visited again. Therefore, its limiting probability is 0. So it makes sense intuitively.
However, I have no idea at all about how I could prove this.
Would appreciate the help
Thanks
I am afraid that this theorem is not fully true without any additional condition.
As known, if a DTMC is irreducible, aperiodic, positive recurrent, it holds that $$ \lim_{n \rightarrow \infty}P_{i,j}^{(n)} = \pi_{j}$$ where $\pi$ denotes the stationary distribution.
Nevertheless, if a DTMC is irreducible and null recurrent, then $ \lim_{n \rightarrow\infty}P_{i,j}^{(n)} = 0 $.