What are stationary distributions?

A stationary distribution is like the favorite spot where things end up after a long time when they’re moving around.

Imagine you have a toy train that goes back and forth between two tracks, Track A and Track B. Sometimes it switches, sometimes it stays on the same track. At first, it might be all over the place, but after a while, you notice it spends about half its time on each track. That balance, where it doesn’t really care which track it’s on anymore, is like a stationary distribution.

A stationary distribution happens when things stop changing much over time. Like how your favorite snack gets eaten the same amount every day, no big surprises, just steady eating!

Think of it as the final answer in a game where everyone has had enough switching and is ready to stay put. The train doesn’t have to be perfectly balanced, but it settles into a pattern that feels right.

So whether you’re tracking trains or anything else moving around, the stationary distribution is just what happens when everything calms down, no more wild switches, just steady, predictable fun!