An elementary proof of existence and uniqueness of stationary distributions for irreducible Markov chains

TL;DR

This paper provides a straightforward, elementary proof of the existence and uniqueness of stationary distributions for irreducible Markov chains, relying solely on basic algebra and the Bolzano-Weierstrass Theorem.

Contribution

It offers a simple, accessible proof of a fundamental Markov chain result, avoiding advanced techniques.

Findings

Existence of a unique stationary distribution for irreducible Markov chains.

The proof uses only basic algebra and the Bolzano-Weierstrass Theorem.

The result applies to matrices with positive or zero entries, row sums equal to 1, and irreducibility.

Abstract

Consider an $n\times n$ matrix $P$ with the following properties. All entries in $P$ are positive or $0$, the sum of each row is 1 and for all $i$ and $j$ in $\{1,\dots,n\}$ there exists a natural number $k$ such that the $(i,j)$ entry of the matrix $P^k$ is strictly positive. Then, there exists a unique row vector $v$ with only strictly positive entries, whose sum of entries is 1 and such that $vP=P$. We present a proof of this well-known result that uses only basic algebra and the Bolzano-Weierstrass Theorem.