Geometric ergodicity under drift and small-set minorization

First assume $V$ is finite-valued. By [F6], choose the bounded-$V$ small subset $C_0$ and ${\lambda }^{\mathrm{\prime }}\in (\lambda ,1)$ with $PV\le {\lambda }^{\mathrm{\prime }}V+b{1}_{C_0}$. Its boundedness is exactly the hypothesis needed for [F8], which now gives $\pi V<\mathrm{\infty }$. Apply the remaining conclusions of [F6] to obtain a skeleton $Q=P^L$, bounded small set $D$, function $h(x,y)=(V(x)+V(y))\mathrm{/}2$, and constants satisfying the hypotheses of [F7]. Since $D$ is bounded in $V$, the residual split expectation is uniformly finite.

Now allow the extended certificate and put $N=\{V=+\mathrm{\infty }\}$ and $E=N^c$. These sets are measurable and $\pi (N)=0$. If $x\in E$, the pointwise drift bound has finite right side, so $PV(x)<\mathrm{\infty }$. Apply [F9] to $V$ under the probability measure $P(x,\cdot )$ to obtain $P(x,N)=0$. Thus $E$ is absorbing, including for every iterate. Also $\varphi (N)=0$: otherwise [F10] would give $\pi (N)>0$. Hence $\varphi (C\cap E)=\varphi (C)>0$, in particular $C\cap E$ is nonempty, and $\varphi {\mathrm{\mid }}_E$ remains a nonzero sigma-finite irreducibility measure.

For every fixed pair $x,y$, [F7] gives $\mathrm{\parallel }{Q}^k(x,\cdot )-Q^k(y,\cdot ){\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}\le M_{0}h(x,y)r_0^k$ with constants independent of the pair. For each measurable event $A$, invariance and the pointwise event bound imply $\mathrm{\mid }{Q}^k(x,A)-\pi (A)\mathrm{\mid }=\mid \int [Q^k(x,A)-Q^k(y,A)]\pi (dy)\mid \le M_0{r}_0^k\int h(x,y)\pi (dy)$. The integrand on the left is a difference of measurable kernel evaluations; no measurable selection of pairwise couplings is needed. By step 1.1, $\int h(x,y)\pi (dy)=\frac{1}{2}(V(x)+\pi V)\le \frac{1}{2}(1+\pi V)V(x)<\mathrm{\infty }$. Taking the event supremum [F5] gives the required $V(x)$-weighted bound at skeleton times.

Restrict the state space to $E$ with its trace sigma-algebra and set $P_E(x,A)=P(x,A)$ for trace-measurable $A\subseteq E$. Absorption makes this a probability kernel, with invariant probability $\pi {\mathrm{\mid }}_E$; a countable generator of $\mathcal{X}$ restricts to a countable generator on $E$. Original irreducibility supplies positive finite-time access to every positive-$\varphi {\mathrm{\mid }}_E$ set from each $x\in E$. For the original $(m,ϵ,\nu )$-minorization on $C$, choose $x\in C\cap E$. Since $P^m(x,N)=0$, minorization gives $\nu (N)=0$, so $C\cap E$ is small for $P_E$ with probability $\nu {\mathrm{\mid }}_E$. Any positive-$\pi {\mathrm{\mid }}_E$ cyclic part for $P_E$ would consist of original measurable subsets of $E$ with the same pointwise transitions under $P$, contradicting the actual aperiodicity definition [F2], which excludes such a part without requiring it to cover the state space. Finally $V{\mathrm{\mid }}_E$ is finite-valued and satisfies the original drift inequality with drift set $C\cap E$.

Write $n=Lk+s$, $0\le s<L$. The finite set of residual powers $P^s$ contracts total variation and the stationary law remains $\pi$. Absorb the finite factor and $r_0^{-s\mathrm{/}L}$ over $s=0,\dots ,L-1$ into a single $M$, and put $r=r_0^{1\mathrm{/}L}$. Then the asserted bound holds for every $n$.

Apply the finite-valued result proved in steps 1.1, 2.1, 3.1 to $(E,P_E,\varphi {\mathrm{\mid }}_E,\pi {\mathrm{\mid }}_E,V{\mathrm{\mid }}_E,C\cap E)$. It gives constants $M<\mathrm{\infty }$ and $0<r<1$ and the weighted bound for every $x\in E$. Both $P^n(x,\cdot )$ and $\pi$ are supported on $E$, so their total variation on the original space equals their total variation on the trace space: every event is replaced by its intersection with $E$. Thus precisely the displayed original-space bound holds on $E$, a full-$\pi$ absorbing set. If $V$ is globally finite, $E=\mathsf{X}$ and the first branch is recovered.