A uniformly ergodic adaptive refresh family

The matrix of $P_{\gamma }$ is $\left(\begin{array}{cc}1-\gamma \mathrm{/}2& \gamma \mathrm{/}2\\ \gamma \mathrm{/}2& 1-\gamma \mathrm{/}2\end{array}\right)\mathrm{.}$ Its entries are continuous in $\gamma$, so the family is jointly measurable on the stated spaces; its rows are probability vectors by $0\le \gamma \le 1$ and [F1]. The matrix is symmetric, and $\pi$ assigns equal mass to both states, so the four joint-flow atoms are symmetric under coordinate swap. Thus $P_{\gamma }$ is reversible by [F2], and $\pi$ is invariant by [F3].

The kernels satisfy $P_{{\gamma }^{\mathrm{\prime }}}(x,\cdot )-P_{\gamma }(x,\cdot )=({\gamma }^{\mathrm{\prime }}-\gamma )(\pi -{\delta }_x)$, so $D_n=\mathrm{\mid }{\mathrm{\Gamma }}_{n+1}-{\mathrm{\Gamma }}_n\mathrm{\mid }\mathrm{/}2\to 0$. This is diminishing adaptation by [F5].

Since ${\mathrm{\Pi }}^2=\mathrm{\Pi }$ and $I\mathrm{\Pi }=\mathrm{\Pi }I=\mathrm{\Pi }$, induction gives $P_{\gamma }^m=(1-\gamma {)}^{m}I+(1-(1-\gamma {)}^m)\mathrm{\Pi }\mathrm{.}$ For either state $x$, $\parallel {\delta }_x-\pi {\parallel }_{\mathrm{T}\mathrm{V}}=1\mathrm{/}2$, so $\parallel P_{\gamma }^m(x,\cdot )-\pi {\parallel }_{\mathrm{T}\mathrm{V}}=(1-\gamma {)}^m\mathrm{/}2\le 2^{-m-1}$ uniformly in $x$ and $\gamma \in [1\mathrm{/}2,1]$. Therefore the family is simultaneously uniformly ergodic.

Steps 2.1 and 1.2 verify the two hypotheses in alternative 1 of [F6]. Hence $\parallel \mathcal{L}(X_n)-\pi {\parallel }_{\mathrm{T}\mathrm{V}}\to 0$ for every initial state.