Uniformized finite-generator stationary residual certificate

Let $Q$ be a finite column generator, let $\nu >0$ satisfy $\nu \ge {\max }_j{r}_j$, and put $P=I+Q\mathrm{/}\nu$. Let $m\ge 1$ and $q=\delta (P^m)<1$. Then the finite master equation $p^{\mathrm{\prime }}=Qp$ has a unique stationary probability $\pi$, characterized among column probability vectors by $Q\pi =0$. For any raw vector $y\in {\mathbb{R}}^n$ and column probability vector $x$, set $c=\mathrm{\parallel }y-x{\mathrm{\parallel }}_1$. Then

$\mathrm{\parallel }y-\pi {\mathrm{\parallel }}_1\le c+\frac{m\mathrm{\parallel }Qx{\mathrm{\parallel }}_1}{\nu (1-q)}\mathrm{.}$

The identity $Qx\mathrm{/}\nu =Px-x$ makes the generator residual directly usable by the discrete certificate. In the exact algorithm branch, $y^{+}=0$ returns candidate-unavailable, and $q\ge 1$ returns certificate-unavailable; neither status supplies an algorithmic accuracy certificate.

In particular, when P and y are rational and the exact branch of Correct and certify a finite-chain stationary candidate returns certified with candidate x and correction c, those outputs satisfy the conditions above.