Multinomial SMC variance unrolls into local contributions

For every integrable $h$ on ${\mathrm{\Theta }}_s$, the Radon–Nikodym identity [F1] gives ${\pi }_{s-1}(E_{s}h)={\pi }_{s-1}(k_s({\upsilon }_{s}h))={\tilde{\pi }}_s({\upsilon }_{s}h)={\pi }_s(h)$. Hence if ${\pi }_s(h)=0$, then $E_{s}h$ has mean zero under ${\pi }_{s-1}$.

For a square-integrable vector function $u$ on ${\mathrm{\Theta }}_s$, expand $u-{\tilde{\pi }}_s(u)=(u-k_{s}u)+(k_{s}u-{\tilde{\pi }}_s(u))\mathrm{.}$ The cross-covariance terms vanish after integrating the first factor conditionally against $k_s$, because $k_s(u-k_{s}u)=0$. Therefore ${\mathrm{Cov}}_{{\tilde{\pi }}_s}(u)=\int {\mathrm{Cov}}_{k_s}(u)(x){\pi }_{s-1}(dx)+{\mathrm{Cov}}_{{\pi }_{s-1}}(k_{s}u)\mathrm{.}$ For $u={\upsilon }_{s}h$ with ${\pi }_s(h)=0$, its ${\tilde{\pi }}_s$ mean is zero by step 1.1. Substituting the multinomial recursion from [F1] thus yields $V_s(h)=V_{s-1}(E_{s}h)+{\tilde{\pi }}_s\text{\hspace{0.17em}⁣}\left[{\upsilon }_s^{2}h{h}^{\mathrm{\top }}\right]\mathrm{.}$

Starting with $h=\bar{\varphi }$ at time $t$, apply step 1.1 to see that each backward iterate $E_{k+1:t}\bar{\varphi }$ has ${\pi }_k$-mean zero. The recursive $\mathrm{\Psi }$ condition also gives its finite target second moment at each such time. Repeated substitution of the one-step identity in step 2.1 gives one new term ${\tilde{\pi }}_k[{\upsilon }_k^2(E_{k+1:t}\bar{\varphi })(E_{k+1:t}\bar{\varphi }{)}^{\mathrm{\top }}]$ at each $k=t,t-1,\dots ,1$, and the remaining initial term is $V_0(E_{1:t}\bar{\varphi })={\mathrm{Cov}}_{{\tilde{\pi }}_0}({\upsilon }_0{E}_{1:t}\bar{\varphi })$. Its mean is zero by step 1.1, so it equals the $k=0$ summand. The same base calculation gives the formula when $t=0$. Each summand is finite: for $k\ge 1$, the recursive $\mathrm{\Phi }$ condition gives a $2+\delta$ proposal moment of ${\upsilon }_k{E}_{k+1:t}\bar{\varphi }$; for $k=0$, the initial ${\mathrm{\Psi }}_0$ moment gives square integrability of ${\upsilon }_0{E}_{1:t}\bar{\varphi }$.