Finite-block perturbation bound for adaptive MCMC

Fix $f:\mathsf{X}\to [0,1]$ measurable. For $i=0,\dots ,N$, set $g_i=P_{{\mathrm{\Gamma }}_n}^{N-i}f$, so $g_N=f$. Joint kernel measurability and [F3] make each $g_i$ measurable in $({\mathrm{\Gamma }}_n,x)$ and $0\le g_i\le 1$. Also $g_i=P_{{\mathrm{\Gamma }}_n}{g}_{i+1}$.

The conditional transition rule [F1] and the identity in step 1.1 give $\mathbb{E}[f(X_{n+N})]-\mathbb{E}[P_{{\mathrm{\Gamma }}_n}^{N}f(X_n)]={\sum }_{i=0}^{N-1}\mathbb{E}\text{\hspace{0.17em}⁣}[(P_{{\mathrm{\Gamma }}_{n+i}}-P_{{\mathrm{\Gamma }}_n})g_{i+1}(X_{n+i})]\mathrm{.}$ Indeed, each summand is the difference between $\mathbb{E}[g_{i+1}(X_{n+i+1})]$ and $\mathbb{E}[g_i(X_{n+i})]$, and these differences telescope from $i=0$ through $N-1$.

By [F2] and the triangle inequality applied pointwise to successive kernels, $\mid (P_{{\mathrm{\Gamma }}_{n+i}}-P_{{\mathrm{\Gamma }}_n})g_{i+1}(X_{n+i})\mid \le {\sum }_{j=0}^{i-1}{D}_{n+j}$. Both sides are measurable, by step 1.1 and the assumed measurability of the $D_m$. Taking expectations bounds the absolute value of the $i$th summand in step 2.1 by ${\sum }_{j=0}^{i-1}\mathbb{E}[D_{n+j}]$. Summing gives the displayed bound for every such $f$; take the supremum using [F2]. For pairwise disjoint $A_{\mathrm{\ell }}$, countable additivity of each $P_{{\mathrm{\Gamma }}_n}^N(X_n,\cdot )$ and [F4] give $Q_{n,N}({\bigcup }_{\mathrm{\ell }}{A}_{\mathrm{\ell }})={\sum }_{\mathrm{\ell }}{Q}_{n,N}(A_{\mathrm{\ell }})$ by monotone convergence; also $Q_{n,N}(\mathsf{X})=1$. Thus $Q_{n,N}$ is a probability measure.