To analyze the Markov chains given their transition probability matrices $P$ and $Q$, we will identify the classes, periodicity, and persistence properties of each chain.

a) Identify the Classes

1. Matrix $P$: 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}$$ - **Classes**: All states communicate with each other: - From state 1 to state 2. - From state 2 to state 3. - From state 3 back to state 1. - Thus, there is **one class**: {1, 2, 3}.
2. Matrix $Q$: 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}$$ - **Classes**: Similarly, all states communicate with each other: - State 1 can go to state 3. - State 3 can go to state 2. - State 2 can go to state 4. - State 4 can go back to state 1. - Thus, there is **one class**: {1, 2, 3, 4}.

b) Periodicity

1. Matrix $P$:

   • All states can be returned to every 3 steps (1 → 2 → 3 → 1).
   • Period: $d = 3$ (periodic).

2. Matrix $Q$:

   • States can be returned every 4 steps (1 → 3 → 2 → 4 → 1).
   • Period: $d = 4$ (periodic).

c) Persistence

1. Matrix $P$:

   • Since it has a period of 3, it is not persistent; states are recurrent but periodic.
   • Class: Persistent null.

2. Matrix $Q$:

   • Similarly, it has a period of 4 and does not return to states in all time frames, making it recurrent but periodic.
   • Class: Persistent null.

Summary:

• P:

  • Classes: {1, 2, 3}
  • Periodicity: Periodic (3)
  • Persistence: Persistent null

• Q:

  • Classes: {1, 2, 3, 4}
  • Periodicity: Periodic (4)
  • Persistence: Persistent null

Questions for Further Exploration:

1. How do the transition probabilities influence the behavior of the Markov chains?
2. What would happen if we modified the transition matrices?
3. Can you provide examples of aperiodic chains?
4. How is persistence affected by the periodicity of a Markov chain?
5. What real-world systems can be modeled using Markov chains similar to these?

Tip:

Understanding the structure of the transition matrix can provide insights into the behavior of the Markov chain, including the classification of states and their long-term behavior.