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Markov chain Monte Carlo estimates expectations under a target probability distribution $\pi$ by simulating a Markov chain that preserves $\pi$. For an integrable observable $h$, the assumptions of a Markov-chain law of large numbers justifyalmost surely. A Gibbs sampler preserves the target through full conditional distributions; a Metropolis–Hastings algorithm proposes a move and accepts or rejects it using a ratio of target and proposal densities. The unknown normalization constant cancels from that ratio, which makes unnormalized Bayesian posterior densities usable.

Successive states are generally dependent, so Monte Carlo precision depends on the observable's integrated autocorrelation time, rather than the raw iteration count alone. Burn-in can reduce initial-state effects but is not a proof of convergence. A correct stationary distribution also does not ensure rapid exploration: isolated modes or strong dependence can produce a long mixing time. Markov chain Monte Carlo convergence diagnostics investigate these limitations.

The summand intended in the question is $d,h(X(t))/S(X(t))$. Under stationarity,Moreover $S(x)\ge d{c}_1$, so the summand is bounded by $1/c_1$. A stationary geometrically ergodic Markov chain satisfies the Markov-chain law of large numbers; hencealmost surely, and therefore in probability. In particular,

Tempered Gibbs sampling combines modified full conditional distributions with state-dependent coordinate selection and importance sampling. For a positive target density $f(x)$, choose strictly positive normalized densities $g_i(x_i\mid x_{-i})$ with compatible support, and defineSelect coordinate $i$ proportionally to $p_i(x)$ and redraw it from $g_i$. The invariant density is $f(x)Z(x)$, not generally $f(x)$. For a move changing only coordinate $i$, stationary transition weights are proportional to $f(x_{-i})g_i(x_i\mid x_{-i})g_i(y_i\mid x_{-i})$, symmetric in old and new values. Integrating also shows $\int fZ=1$.

Under the appropriate Markov-chain law of large numbers, weighted averages with weights $1/Z(x)$ recover expectations under $f$. Flattening a conditional can encourage longer moves, but omitting the weights changes the estimand. The $g_i$ need not be full conditionals of a common second joint distribution.^[ref]