= Product stationarity under single-coordinate Markov updates
{title2=$\pi^{\otimes m}$}
Suppose each coordinate has <stationary distribution> $\pi$ for the same <Markov kernel>. Updating one coordinate using independent new randomness while leaving the others fixed preserves the product distribution $\pi^{\otimes m}$. For a two-state update with $P(1\to1)=a$ and $P(0\to1)=b$, its stationary occupied probability is $p=b/(1-a+b)$. Independent occupied-state indicators then have total mean $mp$ and variance $mp(1-p)$. The conclusion requires the update selection and transition randomness not to introduce dependence on the other coordinates' states.
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