Conditional mutation and weighted correction CLT

At $t=0$, the centered weighted numerator $Z={\upsilon }_0(\varphi -{\pi }_0\varphi )$ has mean zero and finite second moment under the weighted-only assumptions; the ${\mathrm{\Psi }}_0$ assumptions are sufficient for them. If its variance ${\sigma }_Z^2$ is positive, its IID row satisfies Lindeberg because ${\sigma }_Z^{-2}E[Z^2{\mathbf{1}}_{\{\mathrm{\mid }Z\mathrm{\mid }>a{\sigma }_Z\sqrt{N}\}}]\to 0$ by [F7] for each $a>0$. Thus [F4] gives the IID CLT; if its variance is zero, [F6] gives convergence of its centered sample mean to zero. The denominator ${\eta }_0^N({\upsilon }_0)$ converges to one by [F3], since ${\tilde{\pi }}_0({\upsilon }_0)=1$. Slutsky [F5] gives the ratio CLT with variance $V_0(\varphi )$. For $p\ge 1$, $\mathrm{\mid }ax+by{\mathrm{\mid }}^p\le 2^{p-1}(\mathrm{\mid }a{\mathrm{\mid }}^p\mathrm{\mid }x{\mathrm{\mid }}^p+\mathrm{\mid }b{\mathrm{\mid }}^p\mathrm{\mid }y{\mathrm{\mid }}^p)$; choosing the smaller available positive $\delta$ therefore proves closure of ${\mathrm{\Phi }}_t$ under finite linear combinations, by induction from ${\mathrm{\Phi }}_0$. Kernel integration is linear on these integrable functions. The same inequality gives closure of the target and centered weighted $L^2$ conditions in ${\mathrm{\Psi }}_t$, and its centered images combine linearly; induction proves closure of ${\mathrm{\Psi }}_t$. If $\varphi \in {\mathrm{\Psi }}_t$, then $\psi ={\upsilon }_t(\varphi -{\pi }_t\varphi )$ has a finite proposal $2+\delta$ moment because $\varphi ,1\in {\mathrm{\Phi }}_t$, and $k_t\psi =E_t(\varphi -{\pi }_t\varphi )\in {\mathrm{\Psi }}_{t-1}$ by the recursive ${\mathrm{\Psi }}_t$ condition.

For the mutation part, $k_t\mathrm{\mid }\psi {\mathrm{\mid }}^{2+\delta }$ is ${\pi }_{t-1}$-integrable, with integral ${\tilde{\pi }}_t(\mathrm{\mid }\psi {\mathrm{\mid }}^{2+\delta })$. By [F3], $H_N={\check{\eta }}_{t-1}^N(k_t\mathrm{\mid }\psi {\mathrm{\mid }}^{2+\delta })$ converges in probability to this finite value, so $H_N<\mathrm{\infty }$ with probability tending to one. On this event condition on the selected parent paths ${\check{X}}_{t-1}^{1:N}$ and set $m_i=k_t\psi ({\check{X}}_{t-1}^i)$. Each parent has a finite conditional $2+\delta$ moment; under the product kernel in [F2], $D_{i,N}=\psi (X_t^i)-m_i$ are independent and centered. Their row variance is $s_N^2=\frac{1}{N}{\sum }_{i=1}^N{\mathrm{Var}}_{k_t({\check{X}}_{t-1}^i)}(\psi ),$ which converges in probability to ${\sigma }^2={\pi }_{t-1}({\mathrm{Var}}_{k_t}(\psi ))$ by [F3], since ${\mathrm{Var}}_{k_t}(\psi )\le k_t({\psi }^2)$ is integrable by Jensen's inequality. The event removed has probability tending to zero and does not affect the distributional limits below. When ${\sigma }^2>0$, normalize the array by $s_N\sqrt{N}$, where $s_N^2$ is the displayed variance. For every $a>0$, the resulting conditional Lindeberg sum is at most $C_{\delta }{a}^{-\delta }{N}^{-\delta \mathrm{/}2}{s}_N^{-(2+\delta )}{H}_N,$ where $C_{\delta }$ is a finite constant from $\mathrm{\mid }a-b{\mathrm{\mid }}^{2+\delta }\le 2^{1+\delta }(\mathrm{\mid }a{\mathrm{\mid }}^{2+\delta }+\mathrm{\mid }b{\mathrm{\mid }}^{2+\delta })$ and Jensen's inequality. Since $s_N\to \sigma >0$ and $H_N$ converges in probability to a finite value, this conditional Lindeberg sum tends to zero.

Along any subsequence, choose a further subsequence on which $s_N\to \sigma$ almost surely. If $\sigma >0$, a further subsequence makes the normalized Lindeberg sums converge almost surely; [F4] gives a standard-normal limit for the normalized row, and Slutsky [F5] with $s_N\to \sigma$ gives the conditional characteristic-function limit $e^{-u^2{\sigma }^2\mathrm{/}2}$. If $\sigma =0$, conditional [F6] gives $P\text{\hspace{0.17em}⁣}(\mid N^{-1\mathrm{/}2}{\sum }_i{D}_{i,N}\mid >a|{\check{X}}_{t-1}^{1:N})\le \min (1,s_N^2\mathrm{/}{a}^2)⟶0$ for every $a>0$, yielding the same characteristic-function limit with variance zero. The subsequence criterion therefore gives convergence in probability of the conditional characteristic functions along the full sequence. They are bounded by one, so their $L^1$ differences from that limit also tend to zero.

The conditional mean part is $P_N=\sqrt{N}({\check{\eta }}_{t-1}^N(k_t\psi )-{\pi }_{t-1}(k_t\psi )),$ and has the previous selected-population CLT by the hypothesis of the Statement. The total proposal error is $P_N+N^{-1\mathrm{/}2}{\sum }_i{D}_{i,N}$. Multiplying the conditional characteristic function from step 2.1 by the bounded factor $e^{iu{P}_N}$ and integrating shows that its limit is the product of the previous Gaussian characteristic function and $e^{-u^2{\sigma }^2\mathrm{/}2}$. Hence the two covariance contributions add, giving ${\tilde{V}}_t(\psi )$.

For $\varphi \in {\mathrm{\Psi }}_t$, apply the mutation result to $\psi ={\upsilon }_t(\varphi -{\pi }_t\varphi )$ using step 1.1 and the preceding selected CLT for $k_t\psi \in {\mathrm{\Psi }}_{t-1}$. By [F3], ${\eta }_t^N({\upsilon }_t)\to {\tilde{\pi }}_t({\upsilon }_t)=1$ in probability, and the event of a zero denominator has probability tending to zero. On its complement, $\sqrt{N}({\omega }_t^N(\varphi )-{\pi }_t(\varphi ))=\frac{\sqrt{N}{\eta }_t^N({\upsilon }_t(\varphi -{\pi }_t\varphi ))}{{\eta }_t^N({\upsilon }_t)}\mathrm{.}$ Slutsky [F5] gives the stated limit with covariance ${\tilde{V}}_t({\upsilon }_t(\varphi -{\pi }_t\varphi ))=V_t(\varphi )$. The fallback on a zero empirical weight sum changes the ratio only on an event whose probability tends to zero.