# A Simple Proof of the Monotonicity of the Invariant Distribution for a Discrete Markov Chain

## TL;DR

This paper provides a straightforward proof demonstrating that the invariant distribution of a finite-state discrete Markov chain exhibits monotonicity, addressing a recent open question in the field.

## Contribution

It offers a simple, accessible proof of a previously unresolved question about the monotonicity of invariant distributions in finite Markov chains.

## Key findings

- Confirmed the monotonicity property for invariant distributions
- Provided a simplified proof accessible to a broad audience
- Resolved a recent open question in Markov chain theory

## Abstract

This note presents a simple proof of the monotonicity of the invariant distribution of a discrete Markov chain with a finite state space. This answers a question recently raised by David Siegmund.