Lagmental Vicfred

A Stationary Distribution Is a Left Eigenvector by Vicfred

tags: mathematics, Markov chains, stationary distributions, random walks

Starting a chain in pi leaves its one-time marginal distribution unchanged. This is a compact note, but the quantifiers and hypotheses stay on the page.

Set-up

A discrete-time Markov chain has transition matrix \(P=(p_{ij})\) and forgets the past after conditioning on the present. Matrix powers \(P^n\) give multi-step transition probabilities.

$$ \pi_i\ge0,\qquad\sum_i\pi_i=1 $$

A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.

$$ \pi P=\pi $$

The calculation

A worked instance is useful here because it exposes every index that the compressed statement hides.

$$ P=\begin{pmatrix}1-a&a\\b&1-b\end{pmatrix}\quad\Longrightarrow\quad\pi=\left(\frac b{a+b},\frac a{a+b}\right) $$

What survives abstraction

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad \pi_i\ge0,\qquad\sum_i\pi_i=1,\\[5pt] \mathsf{C}\;&:\quad \pi P=\pi. \end{aligned} $$

The boundary

A stationary distribution need not be unique without irreducibility, and convergence to it can fail without aperiodicity.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \pi P=\pi \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Fri 17 March 2017. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.