2004-83.md

course	course_year	question_number	tags	title	year
Markov Chains	IB	83	IB 2004 Markov Chains	1.I.11H	2004

Let $P=\left(P_{i j}\right)$ be a transition matrix. What does it mean to say that $P$ is (a) irreducible, $(b)$ recurrent?

Suppose that $P$ is irreducible and recurrent and that the state space contains at least two states. Define a new transition matrix $\tilde{P}$ by

$$\tilde{P}{i j}=\left{\begin{array}{lll} 0 & \text { if } & i=j \ \left(1-P{i i}\right)^{-1} P_{i j} & \text { if } & i \neq j \end{array}\right.$$

Prove that $\tilde{P}$ is also irreducible and recurrent.