Containment for adaptive MCMC

Use the adaptive MCMC setup of Diminishing adaptation. For $ϵ>0$, a state $x\in \mathsf{X}$, and a fixed parameter $\gamma \in \mathrm{\Gamma }$, define the convergence time $M_{ϵ}(x,\gamma ):=\inf \{m\in \mathbb{N}:m\ge 1,\parallel P_{\gamma }^m(x,\cdot )-\pi {\parallel }_{\mathrm{T}\mathrm{V}}\le ϵ\},$ with $\inf \mathrm{\varnothing }=+\mathrm{\infty }$. The power $P_{\gamma }^m$ is the $m$-fold composition of the fixed kernel $P_{\gamma }$; it is not the adaptive transition law. Assume $M_{ϵ}(X_n,{\mathrm{\Gamma }}_n)$ is measurable for each $n$ and $ϵ>0$.

For a specified initial probability law ${\mu }_0$ of $(X_0,{\mathrm{\Gamma }}_0)$, the adaptive process has containment if for every $ϵ,\delta >0$ there is an integer $N=N(ϵ,\delta )$ such that ${\mathbb{P}}_{{\mu }_0}\text{\hspace{0.17em}⁣}(M_{ϵ}(X_n,{\mathrm{\Gamma }}_n)\le N)\ge 1-\delta \text{for every }n\ge 0.$ A process initialized at the fixed pair $(x_{\ast },{\gamma }_{\ast })$ is the special case ${\mu }_0={\delta }_{(x_{\ast },{\gamma }_{\ast })}$. Thus containment says that the fixed-kernel convergence times, evaluated at the current state and parameter, form a sequence bounded in probability under the specified initial law. It does not assert a uniform deterministic bound for every possible state and parameter.