In a Markov chain, states are grouped into communicating classes, which are sets of states that communicate with each other. Two states $i$ and $j$ are said to communicate if:

1. You can reach state $j$ from state $i$ (denoted $i \rightarrow j$).
2. You can also reach state $i$ from state $j$ (denoted $j \rightarrow i$).

This means that for two states to be in the same class, there must be a two-way communication between them.

Now, let's examine your scenario:

• State $B$ communicates with State $S$, meaning $B \leftrightarrow S$, which implies $B \rightarrow S$ and $S \rightarrow B$.
• State $S$ can reach the absorbing state $I$, but not vice versa, meaning $S \rightarrow I$ but $I \not\rightarrow S$. Since $I$ is an absorbing state, no state can be reached from $I$, including $S$.
• State $M$ can reach State $S$ but not vice versa, meaning $M \rightarrow S$ but $S \not\rightarrow M$.

For $B$ and $S$ to be in the same class, they must be able to communicate both ways. However, if $B \leftrightarrow S$, then both should belong to the same class. But here’s the problem:

• $S$ can reach the absorbing state $I$, which breaks the symmetry because you cannot come back from $I$ to $S$. This lack of two-way communication for some states complicates the classification.

Thus, based on this analysis, $B$ and $S$ could still be in the same class if they communicate with each other. However, the fact that $S$ can reach an absorbing state $I$ (from which no return is possible) means that $B$ and $S$ might not necessarily stay in a recurrent class together if the chain contains absorbing states. Their class structure might be disrupted because absorbing states affect which states can form recurrent communicating classes.

Clarification:

• $B$ and $S$ could still belong to the same transient class (before reaching $I$), but once $S$ reaches the absorbing state $I$, $S$'s dynamics change, and $B$'s path could diverge. So, the concept of same class holds only as long as both remain in transient behavior and can communicate bidirectionally.

Would you like further details or clarifications?

5 Related Questions:

1. What is an absorbing state in a Markov chain, and how does it affect the classification of states?
2. How are transient and recurrent states defined in Markov chains?
3. What is a communication class, and how does it affect the behavior of a Markov chain?
4. Can a Markov chain have both transient and absorbing states, and how does this impact its long-term behavior?
5. How does the concept of irreducibility apply to Markov chains?

Tip:

Always check whether all the states can communicate with each other (in both directions) to identify which states belong to the same class.