Solution
= Solution
Given the current state $x=(x_1,\ldots,x_d)$, a <Random-scan Gibbs sampler> chooses $I$ uniformly from $\{1,\ldots,d\}$, leaves $x_{-I}$ unchanged, and samples the new coordinate from the complete conditional distribution
$$
X_I'\sim\pi(dx_I\mid x_{-I}).
$$
Its transition kernel is
$$
K(x,dy)=\frac1d\sum_{i=1}^d
\pi(dy_i\mid x_{-i})\,\delta_{x_{-i}}(dy_{-i}),
$$
and it leaves $\pi$ invariant.