Conditional residual resampling central limit theorem away from integer weights

Put ${\bar{u}}_N={\eta }_t^N(u)$ and $b_i^N=r(u(X_t^i)\mathrm{/}{\bar{u}}_N)$ on the event ${\bar{u}}_N>0$, whose probability tends to one by [F3]. If $u=1$ almost everywhere, exclude all earlier zero-weight fallback events; their union has probability tending to zero by [F3] and the fixed horizon. On the remaining event every proposal particle avoids proposal-null sets almost surely: initially particles have the proposal law, and inductively positive-weight selection avoids target-null sets, after which mutation avoids ${\tilde{\pi }}_s$-null sets since ${\tilde{\pi }}_s={\pi }_{s-1}{k}_s$. Thus all sampled $u$ values equal one almost surely there, so residual selection is deterministic. The resampling fluctuation is zero outside an event of probability tending to zero, proving the constant-weight branch. In the other case, for the three tests $a=1,\varphi ,{\varphi }^2$, where ${\pi }_t({\varphi }^2)<\mathrm{\infty }$, one has $\frac{1}{N}{\sum }_i{b}_i^{N}a(X_t^i)\stackrel{P}{\to }{\tilde{\pi }}_t(r(u)a)\mathrm{.}$ To verify this triangular empirical limit, fix $M$ and first restrict to $u\le M$ and $\mathrm{\mid }\varphi \mathrm{\mid }\le M$. The compact $u$ interval meets only finitely many positive-integer discontinuities of $r$. Delete open $\gamma$-neighborhoods of those integers. On the remaining compact set, $r(u\mathrm{/}{\bar{u}}_N)\to r(u)$ uniformly as ${\bar{u}}_N\to 1$, and $\mathrm{\mid }a\mathrm{\mid }\le \max (1,M,M^2)$, so [F3] gives convergence of the restricted averages. Each deleted neighborhood has proposal integral of $\mathrm{\mid }a\mathrm{\mid }$ tending to zero as $\gamma \downarrow 0$: it lies where $u\ge 1\mathrm{/}2$, and ${\tilde{\pi }}_t(u\mathrm{\mid }a\mathrm{\mid })={\pi }_t(\mathrm{\mid }a\mathrm{\mid })<\mathrm{\infty }$; the no-atom condition gives the limit zero. [F3] controls its empirical average for each fixed $\gamma$. For the tails, when ${\bar{u}}_N\in [1\mathrm{/}2,2]$, $b_i^N\le 2u(X_t^i)$; hence the $\mathrm{\mid }\varphi \mathrm{\mid }>M$ contribution is bounded by $2{\eta }_t^N(u\mathrm{\mid }a\mathrm{\mid }{\mathbf{1}}_{\{\mathrm{\mid }\varphi \mathrm{\mid }>M\}})$, whose limit tends to zero by target integrability. On $u>M$, use $b_i^N\le 1$ and ${\tilde{\pi }}_t(\mathrm{\mid }a\mathrm{\mid }{\mathbf{1}}_{\{u>M\}})\le M^{-1}{\pi }_t(\mathrm{\mid }a\mathrm{\mid })$, with [F3] again controlling the empirical average. The corresponding target tails obey the same bounds because $r(u)\le u$ and $r(u)\le 1$. Taking first $N\to \mathrm{\infty }$, then $\gamma \downarrow 0$ and $M\to \mathrm{\infty }$, proves the claim.

Taking $a=1$ in step 1.1 gives $N^{-1}{\sum }_i{b}_i^N\to c$. Here $c>0$: if $c=0$, then $u$ is integer-valued almost surely; the no-positive-integer-atom assumption would force $u=0$ almost surely, contradicting ${\tilde{\pi }}_t(u)=1$. Thus the number $M_N={\sum }_i{b}_i^N$ of residual draws diverges and $M_N\mathrm{/}N\to c$. Their conditional sampling probabilities are $b_i^N\mathrm{/}{M}_N$.

Let $m_N=M_N^{-1}{\sum }_i{b}_i^N\varphi (X_t^i)$. Given the particles, the residual offspring contribution to the centered, $\sqrt{N}$-scaled empirical mean is $N^{-1\mathrm{/}2}{\sum }_{\mathrm{\ell }=1}^{M_N}(\varphi (Y_{\mathrm{\ell }}^N)-m_N)$, with IID conditional draws $Y_{\mathrm{\ell }}^N$ from probabilities $b_i^N\mathrm{/}{M}_N$. By step 1.1 its conditional variance converges in probability to $c\{\frac{{\tilde{\pi }}_t(r(u){\varphi }^2)}{c}-{\left(\frac{{\tilde{\pi }}_t(r(u)\varphi )}{c}\right)}^2\}=R_t(\varphi )\mathrm{.}$ The residual draw mean $m_N$ is bounded in probability by the $a=\varphi$ limit in step 1.1 and $M_N\mathrm{/}N\to c>0$. Fix a threshold $a_0>0$ and fixed $K$, then $R\ge K$. On $\{\mathrm{\mid }{m}_N\mathrm{\mid }\le K,{\bar{u}}_N\in [1\mathrm{/}2,2]\}$, for sufficiently large $N$ with $a_0\sqrt{N}>R+K$, the conditional Lindeberg sum for $N^{-1\mathrm{/}2}{\sum }_{\mathrm{\ell }=1}^{M_N}(\varphi (Y_{\mathrm{\ell }}^N)-m_N)$ is at most $4N^{-1}{\sum }_i{b}_i^N\varphi (X_t^i{)}^2{\mathbf{1}}_{\{\mathrm{\mid }\varphi (X_t^i)\mathrm{\mid }>R\}}$. Since $b_i^N\le 2u(X_t^i)$ on this event, it is at most $8{\eta }_t^N(u{\varphi }^2{\mathbf{1}}_{\{\mathrm{\mid }\varphi \mathrm{\mid }>R\}})$. For fixed $R$, [F3] makes this converge to $8{\tilde{\pi }}_t(u{\varphi }^2{\mathbf{1}}_{\{\mathrm{\mid }\varphi \mathrm{\mid }>R\}})=8{\pi }_t({\varphi }^2{\mathbf{1}}_{\{\mathrm{\mid }\varphi \mathrm{\mid }>R\}})$, which tends to zero as $R\to \mathrm{\infty }$. Choosing $K$, then $R$, then $N$ controls the exceptional mean event and the integrable fixed tail; ${\bar{u}}_N\to 1$ handles the normalization event. Thus conditional Lindeberg holds under target $L^2$, with no target higher moment.

Along any subsequence, choose a further subsequence on which the conditional residual variance converges almost surely. If $R_t(\varphi )>0$, the conditional standard deviation is eventually greater than $\sqrt{R_t(\varphi )}\mathrm{/}2$. Apply the bound in step 3.1 at threshold $a\sqrt{R_t(\varphi )}\mathrm{/}2$ and divide by the conditional variance; this gives the Lindeberg condition after normalizing the residual sum by its conditional standard deviation. Conditional [F4] gives a standard-normal limit, and Slutsky [F6] with the conditional standard deviation converging to $\sqrt{R_t(\varphi )}$ gives the characteristic-function limit $e^{-u_0^2{R}_t(\varphi )\mathrm{/}2}$. If $R_t(\varphi )=0$, conditional [F5] bounds the deviation probability by the conditional variance divided by the squared threshold, giving the same degenerate limit. AC is inherited only through the particle law and conditional CLT supplier [F4]; the finite residual calculation adds no choice. The subsequence criterion gives convergence in probability along the full sequence. These conditional characteristic functions are bounded by one. For arbitrary real $v,u_0$, multiply this conditional characteristic function at $u_0$ by $e^{iv\sqrt{N}({\omega }_t^N(\varphi )-{\pi }_t(\varphi ))}$ and integrate. The limit is $e^{-v^{2}V\mathrm{/}2}{e}^{-u_0^2{R}_t(\varphi )\mathrm{/}2}$, proving joint asymptotic independence and adding $R_t(\varphi )$ to $V$. The deterministic-floor part is centered at its conditional expectation, so the conditional mean of the selected empirical law is exactly ${\omega }_t^N(\varphi )$.