Consider an integer m ≥ 3. You are given a homogeneous Markov chain on a finite state space {1, 2, …, m} with transition probability matrix Q and initial distribution π. Let Im represent the identity matrix of order m and Tm represent the number of time periods before the chain returns to state 'm' starting from 'm'. Also, assume that the Markov chain is irreducible, but not necessarily aperiodic or ergodic. Which of the following statements are necessarily correct?
1
If α is an eigenvalue of Q, then |α| ≤ 1.
2
If we have a stationary distribution vector and a transition matrix Q then, on multiplication, the stationary distribution vector remains unchanged.
3
The sequence {\({Q^{n π}} \) }as n -> infinity gives the stable distribution of the chain.
4
If Q is symmetric, then π is necessarily a uniform distribution over the state space.