Diminishing adaptation

Let $(\mathsf{X},\mathcal{X})$ and $(\mathrm{\Gamma },\mathcal{G})$ be nonempty measurable spaces (Measurable spaces and measurable sets). Let $\pi$ be a probability measure on $(\mathsf{X},\mathcal{X})$, and let $(P_{\gamma }{)}_{\gamma \in \mathrm{\Gamma }}$ be a jointly measurable family of probability kernels on $\mathsf{X}$: for every $A\in \mathcal{X}$, $(\gamma ,x)\mapsto P_{\gamma }(x,A)$ is $\mathcal{G}\otimes \mathcal{X}$-measurable and each $P_{\gamma }$ is a probability kernel (Measure kernel and probability kernel). Assume that $\pi$ is invariant for every fixed kernel, meaning $\pi P_{\gamma }=\pi$ (Invariant and stationary distribution for a Markov kernel).

An adaptive MCMC process is a pair of measurable processes $(X_n,{\mathrm{\Gamma }}_n{)}_{n\ge 0}$ with $X_n\in \mathsf{X}$, ${\mathrm{\Gamma }}_n\in \mathrm{\Gamma }$, and natural filtration ${\mathcal{F}}_n=\sigma (X_0,{\mathrm{\Gamma }}_0,\dots ,X_n,{\mathrm{\Gamma }}_n)$, such that for every $A\in \mathcal{X}$, $\mathbb{P}(X_{n+1}\in A\mid {\mathcal{F}}_n)=P_{{\mathrm{\Gamma }}_n}(X_n,A)\text{a.s.}$ The update rule for ${\mathrm{\Gamma }}_{n+1}$ may depend on the history through time $n+1$; it is required only to make ${\mathrm{\Gamma }}_{n+1}$ a measurable random variable.

For $n\ge 0$, define the adaptation size $D_n:={\sup }_{x\in \mathsf{X}}\parallel P_{{\mathrm{\Gamma }}_{n+1}}(x,\cdot )-P_{{\mathrm{\Gamma }}_n}(x,\cdot ){\parallel }_{\mathrm{T}\mathrm{V}},$ where total variation uses the convention $\parallel \mu -\nu {\parallel }_{\mathrm{T}\mathrm{V}}={\sup }_{A\in \mathcal{X}}\mathrm{\mid }\mu (A)-\nu (A)\mathrm{\mid }$ (Total variation distance for probability laws). Assume each $D_n$ is measurable; it lies in $[0,1]$. The process has diminishing adaptation if $D_n\to 0$ in probability as $n\to \mathrm{\infty }$. This condition does not require ${\mathrm{\Gamma }}_n$ to converge.