Finite-horizon SMC CLT classes and variance recursions

Fix a finite horizon $T$. At each $0\le t\le T$, let ${\pi }_t$ be a probability law on a measurable particle space ${\mathrm{\Theta }}_t$. Let ${\tilde{\pi }}_0$ be an initial proposal with ${\pi }_0\ll {\tilde{\pi }}_0$, and fix ${\upsilon }_0=d{\pi }_0\mathrm{/}d{\tilde{\pi }}_0$. For $1\le t\le T$, let $k_t$ be a probability kernel from ${\mathrm{\Theta }}_{t-1}$ to ${\mathrm{\Theta }}_t$, and put ${\tilde{\pi }}_t(A)={\int }_{{\mathrm{\Theta }}_{t-1}}{k}_t(x,A){\pi }_{t-1}(dx)\mathrm{.}$ Assume ${\pi }_t\ll {\tilde{\pi }}_t$ and fix a nonnegative Radon–Nikodym version ${\upsilon }_t=d{\pi }_t\mathrm{/}d{\tilde{\pi }}_t$, so ${\tilde{\pi }}_t({\upsilon }_t)=1$. The BCS-41 sequential importance-resampling algorithm Sequential importance resampling for Feynman–Kac path targets is this scheme on path spaces when $k_t$ appends a proposal transition. Under the support and measurable versions of Feynman–Kac path Radon–Nikodym weight, its normalized incremental weight is ${\upsilon }_t=(Z_{t-1}\mathrm{/}{Z}_t)G_t{r}_t$ for $t\ge 1$, and ${\upsilon }_0=G_0{\rho }_0\mathrm{/}{Z}_0$ initially. The previous normalized target is already part of ${\tilde{\pi }}_t$; one does not multiply by the full earlier path-weight product again.

For a measurable vector function $\varphi :{\mathrm{\Theta }}_t\to {\mathbb{R}}^d$, write $\mu (\varphi )=\int \varphi d\mu$ and ${\mathrm{Cov}}_{\mu }(\varphi )=\int (\varphi -\mu (\varphi ))(\varphi -\mu (\varphi ){)}^{\mathrm{\top }}d\mu ,$ when the second moment is finite. Also write $k_t\varphi (x)=\int \varphi (z)k_t(x,dz)$ and ${\mathrm{Cov}}_{k_t}(\varphi )(x)=\int (\varphi (z)-k_t\varphi (x))(\varphi (z)-k_t\varphi (x){)}^{\mathrm{\top }}{k}_t(x,dz)\mathrm{.}$

The source recursive test classes are ${\mathrm{\Phi }}_0^{(d)}=\{\varphi :{\tilde{\pi }}_0(\mathrm{\parallel }\varphi {\mathrm{\parallel }}^2)<\mathrm{\infty }\},$ and, for $t\ge 1$, ${\mathrm{\Phi }}_t^{(d)}=\{\varphi :\text{ for some }\delta >0,{\tilde{\pi }}_t(\mathrm{\parallel }{\upsilon }_t\varphi {\mathrm{\parallel }}^{2+\delta })<\mathrm{\infty }\text{ and }x\mapsto k_t({\upsilon }_t\varphi )(x)\in {\mathrm{\Phi }}_{t-1}^{(d)}\}\mathrm{.}$ Assume $1\in {\mathrm{\Phi }}_t^{(1)}$ for every $t$. This is the source theorem's explicit normalizing-weight condition. For the recursive post-resampling CLTs proved on this page, use the stronger classes ${\mathrm{\Psi }}_t^{(d)}$. At $t=0$, let ${\mathrm{\Psi }}_0^{(d)}$ consist of the $\varphi \in {\mathrm{\Phi }}_0^{(d)}$ for which ${\tilde{\pi }}_0(\mathrm{\parallel }{\upsilon }_0(\varphi -{\pi }_0(\varphi )){\mathrm{\parallel }}^2)<\mathrm{\infty }$ and ${\pi }_0(\mathrm{\parallel }\varphi {\mathrm{\parallel }}^2)<\mathrm{\infty }$. The initial normalizing denominator needs only its finite first moment ${\tilde{\pi }}_0({\upsilon }_0)=1$. For $t\ge 1$, let ${\mathrm{\Psi }}_t^{(d)}$ consist of the $\varphi \in {\mathrm{\Phi }}_t^{(d)}$ for which ${\pi }_t(\mathrm{\parallel }\varphi {\mathrm{\parallel }}^2)<\mathrm{\infty }$ and $x\mapsto E_t(\varphi -{\pi }_t(\varphi ))(x)\in {\mathrm{\Psi }}_{t-1}^{(d)}$. Here $E_{t}h=k_t({\upsilon }_{t}h)$ is the operator defined below. Only target second moments are required: fixed-tail truncation supplies the conditional selection Lindeberg condition without a target higher-moment assumption. The recursive target moments supply the offspring variance at every earlier selection used in the induction. The source ${\mathrm{\Phi }}_t$ conditions alone need not imply them. The source condition $1\in {\mathrm{\Phi }}_t$ and these recursive target moments are the stated hypotheses for the CLT induction; no higher initial normalizer moment is required.

Define the mutation covariance operators recursively. At time zero, ${\tilde{V}}_0(\varphi )={\mathrm{Cov}}_{{\tilde{\pi }}_0}(\varphi )\mathrm{.}$ For $t\ge 1$, ${\tilde{V}}_t(\varphi )={\widehat{V}}_{t-1}(k_t\varphi )+\int {\mathrm{Cov}}_{k_t}(\varphi )(x){\pi }_{t-1}(dx)\mathrm{.}$ The weighted-estimator covariance is $V_t(\varphi )={\tilde{V}}_t\text{\hspace{0.17em}⁣}({\upsilon }_t(\varphi -{\pi }_t(\varphi )))\mathrm{.}$ For a measurable vector function $h$ on ${\mathrm{\Theta }}_t$, define the forward operator from ${\mathrm{\Theta }}_{t-1}$ by $E_{t}h(x)=k_t({\upsilon }_{t}h)(x)=\int {\upsilon }_t(z)h(z)k_t(x,dz)\mathrm{.}$ For $k<t$, write $E_{k+1:t}=E_{k+1}\circ \cdots \circ E_t$; set $E_{t+1:t}$ to the identity operator. For multinomial offspring resampling set ${\widehat{V}}_t^{\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{i}}(\varphi )=V_t(\varphi )+{\mathrm{Cov}}_{{\pi }_t}(\varphi )\mathrm{.}$ For residual offspring resampling, let $r(u)=u-\lfloor u\rfloor$. When $c_t={\tilde{\pi }}_t(r({\upsilon }_t))>0$, define the formal source candidate $R_t^{\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{d}}(\varphi )={\tilde{\pi }}_t\text{\hspace{0.17em}⁣}(r({\upsilon }_t)\varphi {\varphi }^{\mathrm{\top }})-\frac{{\tilde{\pi }}_t(r({\upsilon }_t)\varphi ){\tilde{\pi }}_t(r({\upsilon }_t)\varphi {)}^{\mathrm{\top }}}{c_t}\mathrm{.}$ Set $R_t=R_t^{\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{d}}$ and ${\widehat{V}}_t^{\mathrm{r}\mathrm{e}\mathrm{s}}=V_t+R_t$ only when ${\tilde{\pi }}_t({\upsilon }_t\in \{1,2,\dots \})=0$; then $c_t>0$ and the finite-sample floor discontinuities are avoided. If ${\upsilon }_t=1$ ${\tilde{\pi }}_t$-almost everywhere, put $R_t=0$. Residual offspring are deterministic whenever all sampled weights equal one. In SIR this holds almost surely outside earlier zero-weight fallback events, whose probability tends to zero at a fixed horizon; the almost-everywhere condition alone does not prescribe weights on proposal-null states reached after a fallback. No residual covariance is assigned when $c_t=0$ but ${\upsilon }_t$ is not almost everywhere one: the finite-sample normalization can move positive integer weights across floor boundaries. The B-page counterexamples show why the general positive-integer-atom branch remains unassigned.

The classes and recursions are definitions. The fixed-horizon CLT proofs below establish finiteness where their stated hypotheses apply. In particular, the source class ${\mathrm{\Phi }}_t^{(d)}$ alone does not ensure ${\mathrm{Cov}}_{{\pi }_t}(\varphi )<\mathrm{\infty }$; the recursive target-moment class ${\mathrm{\Psi }}_t^{(d)}$ is used for the stated post-resampling conclusions. The residual formula requires a separate continuity condition at positive integer weights; the B-page records finite-state failures without it. For an arbitrary weighted initial proposal, the source class ${\mathrm{\Phi }}_0=L^2({\tilde{\pi }}_0)$ is not by itself enough: the displayed initial weighted-numerator and target moments in ${\mathrm{\Psi }}_0$ are imposed explicitly.

The stronger $\mathrm{\Psi }$ hypotheses retain both locally proved fixed-horizon outputs, not the false claim that every source-$\mathrm{\Phi }$ test has finite post-selection covariance. Separately, a weighted-only initial ratio CLT needs only a target-integrable test and its centered weighted numerator in $L^2({\tilde{\pi }}_0)$; its denominator still needs only ${\tilde{\pi }}_0({\upsilon }_0)=1$.