# Normal distributions of finite Markov chains

## TL;DR

This paper presents a novel way to express the stationary distribution of finite Markov chains as sums of specific normal distributions linked to planar graphs with loops, building on previous algebraic methods.

## Contribution

It introduces a new representation of stationary distributions using normal distributions associated with particular planar graphs, extending prior algebraic approaches.

## Key findings

- Stationary distributions can be expressed as sums of normal distributions.
- Normal distributions are associated with planar graphs with loops.
- The approach builds on semaphore codes and graph expansions.

## Abstract

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.