  We show that the stationary distribution of a finite Markov chain can be
expressed as the sum of certain normal distributions. These normal
distributions are associated to planar graphs consisting of a straight line
with attached loops. The loops touch only at one vertex either of the straight
line or of another attached loop. Our analysis is based on our previous work,
which derives the stationary distribution of a finite Markov chain using
semaphore codes on the Karnofsky--Rhodes and McCammond expansion of the right
Cayley graph of the finite semigroup underlying the Markov chain.
