Diminishing adaptation with loss of containment

Each row of $P_{\gamma }$ is a probability vector, and for fixed $x$ and $A$ the evaluation $P_{\gamma }(x,A)$ is affine in $\gamma$. Hence the family is jointly measurable on the stated finite discrete state space and Borel parameter space. Also $\pi P_{\gamma }=(1-\gamma )\pi +\gamma \pi =\pi$. The same projection identity ${\mathrm{\Pi }}^2=\mathrm{\Pi }$ gives $P_{\gamma }^m(x,\cdot )=(1-\gamma {)}^m{\delta }_x+(1-(1-\gamma {)}^m)\pi ,$ so $\parallel P_{\gamma }^m(x,\cdot )-\pi {\parallel }_{\mathrm{T}\mathrm{V}}=(1-\gamma {)}^m\mathrm{/}2\to 0$ for each fixed $\gamma$. Thus all individual kernels are ergodic for the common invariant target.

The difference of two successive rows is $({\mathrm{\Gamma }}_{n+1}-{\mathrm{\Gamma }}_n)(\pi -{\delta }_x)$, so $D_n=\mathrm{\mid }{\mathrm{\Gamma }}_{n+1}-{\mathrm{\Gamma }}_n\mathrm{\mid }\mathrm{/}2\to 0$.

Let $r_0=1$ and $r_n={\prod }_{j=0}^{n-1}(1-{\mathrm{\Gamma }}_j)$. The law recursion ${\mu }_{n+1}={\mu }_n{P}_{{\mathrm{\Gamma }}_n}$ gives ${\mu }_n=r_n{\delta }_0+(1-r_n)\pi$. Since ${\sum }_j{\mathrm{\Gamma }}_j=1\mathrm{/}2$ and ${\mathrm{\Gamma }}_j\le 1\mathrm{/}4$, the elementary bound $\log (1-u)\ge -2u$ for $0\le u\le 1\mathrm{/}2$ yields $r_n\ge \exp (-2{\sum }_j{\mathrm{\Gamma }}_j)=e^{-1}$. Hence $\parallel {\mu }_n-\pi {\parallel }_{\mathrm{T}\mathrm{V}}=r_n\mathrm{/}2\ge e^{-1}\mathrm{/}2$, so the marginal law does not converge to $\pi$.

For $ϵ=1\mathrm{/}4$, the formula in step 1.1 shows $M_{ϵ}(x,{\mathrm{\Gamma }}_n)\to \mathrm{\infty }$ for both states as ${\mathrm{\Gamma }}_n\to 0$. Thus for every finite $N$, the probability of $M_{ϵ}(X_n,{\mathrm{\Gamma }}_n)\le N$ is eventually zero, contradicting containment in [F5].