 Solution

Let $\bar{\pi }$ be the normalized target and ${\bar{\pi }}_{-i}$ its marginal distribution. A coordinate-$i$ kernel $P_i$ leaves $x_{-i}=x_{-i}^{\prime }$ and has the joint old-new measureThis is symmetric in the old and new state: the common remaining coordinates are fixed and the two conditional factors exchange places. Thus each coordinate kernel satisfies detailed balance. Averaging them with the fixed weights $1/p$ provesIntegrating out the old state proves invariance of $\bar{\pi }$. The common normalization cancels, so detailed balance can also be written with the unnormalized target. This establishes detailed balance of a random-scan Gibbs sampler and a reversible Markov chain. Fresh independent random choices make the update a Markov chain, as explained above; invariance is not an assertion that an arbitrarily initialized chain starts in stationarity.