Any two-state chain

Topic: Markov chains · more questions on this topic
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A chain on two states moves from state 1 to state 2 with probability a and from state 2 to state 1 with probability b, where 0<a,b<1. Show that its limiting distribution is (b,a)/(a+b).

Worked answer (click to show)

P=\begin{pmatrix}1-a&a\\b&1-b\end{pmatrix}.

The first balance equation is \pi_1=(1-a)\pi_1+b\pi_2, that is a\pi_1=b\pi_2: in the long run, the flow from 1 to 2 equals the flow back. With \pi_1+\pi_2=1, \pi_1=b/(a+b) and \pi_2=a/(a+b). Because 0<a,b<1, the chain is irreducible and P_{11}>0 makes it aperiodic, so this stationary vector is the limit.