A bounded-weight mixing filter has bounded asymptotic variance

Every transition row is $(1\mathrm{/}2,1\mathrm{/}2)$, so the target and proposal transitions coincide and the likelihood contributes no correction. By [F1], ${\upsilon }_s=1$, hence $B=1$, and ${\pi }_s$ is uniform for every $s\ge 0$. Since the row does not depend on its predecessor, [F2] gives $\delta (K)=0$; its density rows are identical, so [F3] also applies with $C=1$.

On the closed last-state subspace the forward operator is $E_s{h}_s=K{h}_s$, which maps a last-state test to its uniform average. The true transition and proposal density versions relative to counting measure are all $1\mathrm{/}2$, so [F4] applies with $C=1$, $A=1$, and $\rho =0$. No contraction of arbitrary path-history functions is asserted. The test has $\mathrm{\Delta }\varphi =1$ and is centered by $1\mathrm{/}2$. Every earlier backward image is zero; the final $k=t$ term in the variance expansion is ${\mathrm{Var}}_{{\pi }_t}(\varphi )=1\mathrm{/}4$. Hence $V_t(\varphi )=1\mathrm{/}4$ for every $t$.

Applying [F5] with $A=B=1$, $\rho =0$ and $\mathrm{\Delta }\varphi =1$ gives $V_t(\varphi )\le 1$, uniformly in $t$; the exact variance $1\mathrm{/}4$ from step 1.2 satisfies the bound. This verifies a time-uniform asymptotic-variance result under explicit mixing and weight conditions.