91Ó°ÊÓ

Specify the classes of the following Markov chains, and determine whether they are transient or recurrent: $$ \begin{aligned} &\mathbf{P}_{1}=\left|\begin{array}{ccc} 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & 0 \end{array}\right| & \mathbf{P}_{2}=\left|\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 1 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right| \\ &\mathbf{P}_{3}=\left\|\begin{array}{lllll} \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 \\ \frac{1}{4} & \frac{1}{2} & \frac{1}{4} & 0 & 0 \\ \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} \\ 0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} \end{array}\right\| & \mathbf{P}_{4}=\mid \begin{array}{lllll} \frac{1}{2} & \frac{1}{4} & 0 & 0 & 0 \\ \frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & \vdots & 3 & 0 \\ 1 & 0 & 0 & 0 & 0 \end{array} \| \end{aligned} $$

Step by step solution

01

We can see that every state in \(\mathbf{P}_1\) can be reached from any other state in one step, so the Markov chain is irreducible. There is only one class in this case. Since the chain is irreducible, we can check the aperiodicity using any state, say state 0. The least common multiple of the numbers of steps needed to go from state 0 to state 0 is 2, so the chain is aperiodic. To determine if the states are transient or recurrent, we'll use the criterion mentioned above. \[ \sum_{n=0}^{\infty} p_{00}^{(2n)} = \sum_{n=0}^{\infty} (1)^n = \infty \] Since the sum is infinite, all the states in this Markov Chain are recurrent. #Step 2: Analyze \(\mathbf{P}_2\)

In \(\mathbf{P}_2\), the transition between states varies depending on the current state. State 0 and 1 are communicating with only state 3. State 2 is communicating with states 0, 1, and 3. Therefore, there are 2 classes: {0, 1, 3} and {2}. Since there is more than one class, the chain is reducible. We'll now analyze the aperiodicity in the first class, {0, 1, 3}. The least common multiple of the number of steps required to go from state 0 to state 0 is infinite, so this class is aperiodic. To check the transient or recurrent nature of states, since state 2 is communicating with the first class, it is a transient state. Note that, if a class has a transient state, then all other states in that class are also transient. Since states 0, 1, and 3 are not communicating with any of the other states, they are recurrent. #Step 3: Analyze \(\mathbf{P}_3\)

02

In \(\mathbf{P}_3\), the classes are very easy to observe: {0, 1, 2} and {3, 4}. Both classes are irreducible. The Markov Chain is reducible. Both classes are easily shown to be aperiodic. We can determine the transient or recurrent nature of states by inspection: The first class {0, 1, 2} will be recurrent as the system tends to stay in this class with non-zero probability. The second class {3, 4} will also be recurrent as both states have a non-zero probability of staying in the same state. #Step 4: Analyze \(\mathbf{P}_4\)

In \(\mathbf{P}_4\), states 0 and 1 are only communicating with each other. State 2 is an absorbing state (once the system gets into state 2, it stays there forever). State 3 is reaching only state 4, and state 4 is reaching only state 0. So, there are 3 classes: {0, 1}, {2}, {3, 4}. The Markov chain is reducible. States 0 and 1 are communicating with each other and are aperiodic. State 2 is an absorbing state (aperiodic, too). States 3 and 4 are also aperiodic. In terms of recurrence, since states 3 and 4 are reaching state 0 with non-zero probability, they are transient. State 2 is absorbing, so it is recurrent. States 0 and 1 will also be recurrent as they continue to communicate with each other with non-zero probabilities.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Irreducible Markov Chains

An irreducible Markov chain is one in which it's possible to get from any state to any other state in a finite number of steps. In simple terms, no matter where you start, you can potentially reach every other state. This property means there is only one class in the Markov chain. Since every state is interconnected, we say the chain is strongly connected.

When looking at a Markov Chain matrix like \(\mathbf{P}_1\) from the exercise, we can see that each state communicates with every other state, making it irreducible. This is useful because it simplifies many analyses, particularly when determining the nature of all states being recurrent or otherwise.

Reducible Markov Chains

A reducible Markov chain has at least two separate classes of states, meaning there exist states that cannot be reached from certain other states. This lack of full connectivity categorizes states into groups, or "classes."

For example, if you scrutinize \(\mathbf{P}_2\), you'll note that some states can only transition to specific others, creating distinct classes that don't communicate. The chain is divided into these separate sets of states, which can either be further analyzed as for their transient or recurrent nature.

Transient States

Transient states are those where it is possible to leave and never return with certainty. Think of them as temporary stops where, eventually, over time, the process will move on and not come back.

To identify transient states, see if there’s a non-zero probability that, starting from that state, the chain moves to another set of states permanently. For \(\mathbf{P}_2\), for instance, state 2 is transient because once the process leaves it, there is no mechanism in it for mandatory return.

Recurrent States

Recurrent states are ones you can return to eventually, for sure, given that you have started in it. There is always a non-zero probability of coming back to a recurrent state at some point because it's a part of a closed communication class.

Taking \(\mathbf{P}_3\) as an example, all states are recurrent because once the system reaches any state in its classes, there is a strong probability that it'll circle back around to that state again. Recurrent states essentially trap the process within their class over time.

Aperiodic Markov Chains

Aperiodicity in Markov chains means there isn't a fixed cycle or period within which a state must return to its starting point. Essentially, the state can return after varied numbers of steps each time.

An example would be when analyzing \(\mathbf{P}_1\), where the transitions do not restrict to specific periodic patterns. The least common multiple of step cycles for any state is "1," defining the chain as aperiodic. This indicates more flexibility in when state revisits can happen.