To solve this problem, let's go through each question for the Markov chains defined by matrices $P$ and $Q$.

1. Transition Matrices

Given: $P = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}, \quad Q = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}$

Analysis of Matrix $P$

(a) Identify the class

The matrix $P$ represents a cyclic transition between states:

• State 1 transitions to State 2.
• State 2 transitions to State 3.
• State 3 transitions back to State 1.

Since these transitions form a closed loop, all states (1, 2, and 3) belong to the same class.

(b) Check for periodicity

The period of each state in this cycle is 3 (as it takes three steps to return to the starting state). Therefore, this Markov chain is periodic with period 3.

(c) Identify persistent null or persistent non-null

Since all states are in a single closed class with a periodic cycle, each state is persistent non-null (also known as recurrent), meaning that starting from any state, the system will always return to that state in finite time.

Analysis of Matrix $Q$

(a) Identify the class

For matrix $Q$, let's examine the transitions:

• State 1 transitions to State 2.
• State 2 transitions to State 3.
• State 3 has no outgoing transitions.
• State 4 transitions to State 1.

This structure reveals two classes:

• Class 1: Contains states 1, 2, and 3. These states communicate among each other, but state 3 is an absorbing state as it has no transitions out.
• Class 2: Contains only state 4, which transitions to state 1.

(b) Check for periodicity

• Class 1 (states 1, 2, and 3) has no periodic cycle because state 3 is absorbing. Once the system reaches state 3, it cannot leave. Hence, Class 1 is aperiodic.
• Class 2 (state 4) transitions to state 1 and hence does not form a cycle with other states. This class is also aperiodic.

(c) Identify persistent null or persistent non-null

• Class 1: Since state 3 is an absorbing state, states 1 and 2 are transient, while state 3 is persistent null (since once reached, it cannot transition elsewhere).
• Class 2: State 4 is persistent non-null as it continues the cycle by transitioning to state 1 repeatedly, indirectly leading to a recurring pattern in the system.

Summary of Results

For matrix $P$:

• Class: {1, 2, 3}
• Periodicity: Periodic (period 3)
• Persistence: Persistent non-null

For matrix $Q$:

• Classes:
  • Class 1: {1, 2, 3} (with state 3 as absorbing)
  • Class 2: {4}
• Periodicity:
  • Class 1: Aperiodic
  • Class 2: Aperiodic
• Persistence:
  • Class 1: State 3 is persistent null; states 1 and 2 are transient.
  • Class 2: Persistent non-null.

Would you like further details on any part of this solution, or do you have any specific questions?

Additional Related Questions:

1. How do we compute the period of a state in a Markov chain?
2. What is the significance of an absorbing state in a Markov chain?
3. How do transient and persistent states affect the long-term behavior of a Markov chain?
4. What are the conditions for a Markov chain to be classified as periodic?
5. How do classes in a Markov chain help in understanding its structure?

Tip:

For complex Markov chains, breaking down the transition matrix into smaller classes or components can simplify analysis and clarify the overall structure.