General-state Markov-chain convergence under recurrence conditions

Apply [F6] under the aperiodicity and invariance hypotheses [F1] and [F3]. This gives convergence on a set $G$ with $\pi (G)=1$. The set is measurable because [F7] expresses total variation as a supremum over the countable generating algebra.

Assume Harris recurrence and let $N$ be any $\pi$-null measurable set. If $\varphi (N)>0$, [F2] says every start hits $N$ almost surely. Averaging over $\pi$ would give stationary probability one to ever hitting $N$, but stationarity gives ${\mathrm{Pr}}_{\pi }(X_n\in N)=\pi (N)=0$ for each $n$, and a countable union gives probability zero. Therefore every $\pi$-null set is $\varphi$-null. In particular $\varphi (G^c)=0$, so $\varphi (G)>0$.

Harris recurrence [F2] implies every start hits $G$ at a finite time $T$ almost surely. For fixed $x$, split the time-$n$ law according to $T>n\mathrm{/}2$ and $T\le n\mathrm{/}2$. The first contribution is at most ${\mathrm{Pr}}_x(T>n\mathrm{/}2)\to 0$.

On $T\le n\mathrm{/}2$, the strong Markov property [F8] gives conditional law $P^{n-T}(X_T,\cdot )$, with $X_T\in G$. For each fixed $t$, let ${\lambda }_t$ be the finite law of $X_T$ on $\{T=t\}$. For any measurable event $A$, conditioning on $T$ and $X_T$ bounds the absolute event-probability difference from $\pi (A)$ by ${\mathrm{Pr}}_x(T>n\mathrm{/}2)+{\sum }_{t\le n\mathrm{/}2}{\int }_G\mathrm{\parallel }{P}^{n-t}(y,\cdot )-\pi {\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}{\lambda }_t(dy)\mathrm{.}$ Taking the supremum over $A$ gives the same bound for total variation. For fixed $K$ and $n\ge 2K$, the terms with $t\le K$ tend to zero by [F10]: the distance functions are measurable by [F7], converge pointwise on $G$, and are bounded by one; the terms with $K<t\le n\mathrm{/}2$ are at most ${\mathrm{Pr}}_x(T>K)$. Also ${\mathrm{Pr}}_x(T>n\mathrm{/}2)\to 0$. Letting first $n\to \mathrm{\infty }$ and then $K\to \mathrm{\infty }$ proves convergence. The path law is supplied by [F9] under [A1].