MMiDS 7.4: Self-Assessment Quiz

Which of the following is a sufficient condition for a Markov chain to be aperiodic?

• a) The chain is irreducible.
• b) The chain has a unique stationary distribution.
• c) The chain is lazy.
• d) The chain has a finite state space.

In a Markov chain with state space \( S \) and transition matrix \( P \), what does it mean if the chain is lazy?

• a) \( \mathbb{E}[X_t] = 0 \) for all \( t \).
• b) \( P_{ii} > 0 \) for all \( i \in S \).
• c) \( P_{ij} = 0 \) for all \( i \ne j \).
• d) \( \mathrm{Var}(X_t) = 1 \) for all \( t \).

The Convergence to Equilibrium Theorem applies to which type of Markov chains?

• a) Irreducible and aperiodic chains
• b) Irreducible and periodic chains
• c) Reducible and aperiodic chains
• d) Reducible and periodic chains

The Ergodic Theorem applies to which type of Markov chains?

• a) Irreducible chains
• b) Aperiodic chains
• c) Reducible chains
• d) Periodic chains

Which of the following describes a key difference between the Convergence to Equilibrium Theorem and the Ergodic Theorem?

• a) The Convergence to Equilibrium Theorem applies to periodic chains, while the Ergodic Theorem applies to aperiodic chains.
• b) The Convergence to Equilibrium Theorem concerns the state distribution, while the Ergodic Theorem concerns the frequency of state visits.
• c) The Convergence to Equilibrium Theorem requires a diagonal transition matrix, while the Ergodic Theorem does not.
• d) The Convergence to Equilibrium Theorem is applicable only to finite Markov chains, while the Ergodic Theorem is not.

In the context of the proof of the Convergence to Equilibrium Theorem, what is the coupling time \( T \)?

• a) The time at which the joint process \( ( (X_t, Y_t) )_{t \geq 0} \) becomes a Markov chain
• b) The first time that \( X_T = Y_T \)
• c) The time at which the distributions of \( X_t \) and \( Y_t \) converge to the stationary distribution
• d) The time at which the marginal processes \( X_t \) and \( Y_t \) become independent