Proving that a HMC state is recurrent or transient?

Question

Looking at the HMC $$\begin{bmatrix} 1-\alpha & \alpha \\ 0 & 1 \end{bmatrix} $$

How do I prove that the state 2 is recurrent and that state 1 is transient? What does it actually mean by state 1 and state 2?

Thanks for any guidance

Answer 1

State 1 and state 2 can be anything, for example state 1 = day is sunny and state 2 = day is rainy. Then your matrix tells you the probability of going from sunny to rainy is $\alpha$.

As there is only two states it is enough to show state 1 is transient, as then necessarily the system will always be in state 2. By definition a state is transient if there is a positive probability that the system will leave that state and never return. By looking at your matrix, we see that if the system is in state 2 (a rainy day), the probability of it being a rainy day tomorrow is 1, i.e. guarenteed. Also there is a positive probability of leaving state 1. Hence the result.