2005-77.md

course	course_year	question_number	tags	title	year
Markov Chains	IB	77	IB 2005 Markov Chains	3.I.9D	2005

Prove that if two states of a Markov chain communicate then they have the same period.

Consider a Markov chain with state space ${1,2, \ldots, 7}$ and transition probabilities determined by the matrix

$$\left(\begin{array}{ccccccc} 0 & \frac{1}{4} & \frac{1}{4} & 0 & 0 & \frac{1}{4} & \frac{1}{4} \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & \frac{1}{3} & 0 & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{2} & 0 & 0 & 0 & 0 & \frac{1}{2} & 0 \\ \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & 0 & \frac{1}{6} & \frac{1}{6} \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \end{array}\right)$$

Identify the communicating classes of the chain and for each class state whether it is open or closed and determine its period.