= Three-level no claims discount equilibrium
{title2=$\pi\propto(p_2p_3,(1-p_1)p_3,(1-p_1)(1-p_2))$}
For a three-level <birth-death chain> whose down-or-stay claim probabilities are $p_1,p_2,p_3$, with each in $(0,1)$, <detailed balance> is $\pi_1(1-p_1)=\pi_2p_2$ and $\pi_2(1-p_2)=\pi_3p_3$. Normalizing gives the displayed <stationary distribution>. The finite chain is irreducible and has endpoint holding probabilities, so it is aperiodic and its level proportions converge to this distribution from any initial level.
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