Common drift and minorization give containment under diminishing adaptation, strong aperiodicity, and a finite initial V-moment

For each $r$, the conditional transition rule [F1] and the drift inequality [F3] give $\mathbb{E}[(V\wedge r)(X_{n+1})\mid {\mathcal{F}}_n]\le \lambda V(X_n)+b$. Take expectations and let $r\to \mathrm{\infty }$ by [F6]; assuming $\mathbb{E}V(X_n)<\mathrm{\infty }$, this gives $\mathbb{E}V(X_{n+1})\le \lambda \mathbb{E}V(X_n)+b$. Starting from the finite initial moment and iterating yields ${\sup }_{n\ge 0}\mathbb{E}V(X_n)\le \mathbb{E}V(X_0)+\frac{b}{1-\lambda }=:K<\mathrm{\infty }\mathrm{.}$

Fix $ϵ,\delta >0$. Choose $R\ge K\mathrm{/}\delta$ and then an integer $N\ge 1$ such that $MR{\rho }^N\le ϵ$. Markov's inequality [F7] and the moment bound in step 1.1 give $\mathbb{P}(V(X_n)>R)\le \delta$ for every $n$. On the complementary event [F4] gives $\parallel P_{{\mathrm{\Gamma }}_n}^N(X_n,\cdot )-\pi {\parallel }_{\mathrm{T}\mathrm{V}}\le ϵ$, so $M_{ϵ}(X_n,{\mathrm{\Gamma }}_n)\le N$. This proves containment as defined in [F2].

Diminishing adaptation is [F1], and containment is step 2.1. The containment branch of [F5] therefore gives $\parallel \mathcal{L}(X_n)-\pi {\parallel }_{\mathrm{T}\mathrm{V}}\to 0$.