[proofplan]
We prove the stronger assertion that, at every time $s\leq t$, both the pre-weighting empirical measure $\xi_{s,N}$ and the post-weighting empirical estimate $\eta_{s,N}$ are consistent against every bounded measurable [test function](/page/Test%20Function). The only analytic point is that normalization is continuous in probability when the limiting denominator is strictly positive; we prove this ratio step directly, including the event where the empirical denominator may vanish. The induction then alternates between the assumed resampling-and-mutation consistency property and this normalization argument. Taking $s=t$ and the test function $h$ gives the theorem.
[/proofplan]
[step:Prove that ratios converge when the limiting denominator is positive]
We first record the elementary normalization fact used at every time. Let $(X_N)_{N\in\mathbb{N}}$ and $(Y_N)_{N\in\mathbb{N}}$ be real-valued random variables on $(\Omega,\mathcal{F},\mathbb{P})$, and let $x\in\mathbb{R}$ and $y>0$ be constants such that $X_N\xrightarrow{\mathbb{P}}x$ and $Y_N\xrightarrow{\mathbb{P}}y$. Define $R_N$ by $R_N=X_N/Y_N$ on the event $\{Y_N>0\}$ and by an arbitrary fixed real value on the event $\{Y_N\leq 0\}$. Then $R_N\xrightarrow{\mathbb{P}}x/y$.
Indeed, since $y>0$,
\begin{align*}
\mathbb{P}(Y_N\leq y/2)\leq \mathbb{P}(|Y_N-y|\geq y/2)\to 0.
\end{align*}
On the event $\{Y_N>y/2\}$, we estimate
\begin{align*}
\left|\frac{X_N}{Y_N}-\frac{x}{y}\right|\leq \frac{|X_N-x|}{Y_N}+\frac{|x|\,|Y_N-y|}{yY_N}\leq \frac{2}{y}|X_N-x|+\frac{2|x|}{y^2}|Y_N-y|.
\end{align*}
For any $\varepsilon>0$, choose positive constants $\varepsilon_1$ and $\varepsilon_2$ such that
\begin{align*}
\frac{2}{y}\varepsilon_1+\frac{2|x|}{y^2}\varepsilon_2<\varepsilon.
\end{align*}
Then
\begin{align*}
\mathbb{P}\left(\left|R_N-\frac{x}{y}\right|>\varepsilon\right)\leq \mathbb{P}(Y_N\leq y/2)+\mathbb{P}(|X_N-x|>\varepsilon_1)+\mathbb{P}(|Y_N-y|>\varepsilon_2),
\end{align*}
and the right-hand side tends to $0$. Hence $R_N\xrightarrow{\mathbb{P}}x/y$.
[guided]
The point of this step is to handle normalization without hiding the possible finite-$N$ event where the empirical denominator is zero. We prove the required ratio convergence directly.
Let $(X_N)_{N\in\mathbb{N}}$ and $(Y_N)_{N\in\mathbb{N}}$ be real-valued random variables on $(\Omega,\mathcal{F},\mathbb{P})$, and let $x\in\mathbb{R}$ and $y>0$ be constants with $X_N\xrightarrow{\mathbb{P}}x$ and $Y_N\xrightarrow{\mathbb{P}}y$. Define $R_N$ by $R_N=X_N/Y_N$ on $\{Y_N>0\}$ and by any fixed real value on $\{Y_N\leq 0\}$. Since the limit $y$ is strictly positive, the denominator is eventually positive with probability tending to one:
\begin{align*}
\mathbb{P}(Y_N\leq y/2)\leq \mathbb{P}(|Y_N-y|\geq y/2)\to 0.
\end{align*}
On the event $\{Y_N>y/2\}$, the denominator is bounded away from zero, so the ratio can be estimated algebraically:
\begin{align*}
\left|\frac{X_N}{Y_N}-\frac{x}{y}\right|=\left|\frac{yX_N-xY_N}{yY_N}\right|.
\end{align*}
Add and subtract $xy$ in the numerator and use the triangle inequality:
\begin{align*}
\left|\frac{yX_N-xY_N}{yY_N}\right|\leq \frac{|X_N-x|}{Y_N}+\frac{|x|\,|Y_N-y|}{yY_N}.
\end{align*}
Because $Y_N>y/2$ on the event under consideration, this gives
\begin{align*}
\left|\frac{X_N}{Y_N}-\frac{x}{y}\right|\leq \frac{2}{y}|X_N-x|+\frac{2|x|}{y^2}|Y_N-y|.
\end{align*}
Now fix $\varepsilon>0$. Choose $\varepsilon_1>0$ and $\varepsilon_2>0$ such that
\begin{align*}
\frac{2}{y}\varepsilon_1+\frac{2|x|}{y^2}\varepsilon_2<\varepsilon.
\end{align*}
If $Y_N>y/2$, $|X_N-x|\leq\varepsilon_1$, and $|Y_N-y|\leq\varepsilon_2$, then the preceding estimate implies $|R_N-x/y|\leq\varepsilon$. Therefore
\begin{align*}
\mathbb{P}\left(\left|R_N-\frac{x}{y}\right|>\varepsilon\right)\leq \mathbb{P}(Y_N\leq y/2)+\mathbb{P}(|X_N-x|>\varepsilon_1)+\mathbb{P}(|Y_N-y|>\varepsilon_2).
\end{align*}
Each term on the right tends to $0$ by convergence in probability of $Y_N$ and $X_N$. Hence $R_N\xrightarrow{\mathbb{P}}x/y$.
[/guided]
[/step]
[step:Initialize the induction at time $0$]
We prove by induction on $s\in\{0,\dots,t\}$ that for every bounded measurable $\varphi:E_s\to\mathbb{R}$,
\begin{align*}
\xi_{s,N}(\varphi)\xrightarrow{\mathbb{P}}\xi_s(\varphi)
\end{align*}
and
\begin{align*}
\eta_{s,N}(\varphi)\xrightarrow{\mathbb{P}}\eta_s(\varphi).
\end{align*}
For $s=0$, the first convergence is exactly the assumed consistency of $\xi_{0,N}$. Let $\varphi:E_0\to\mathbb{R}$ be bounded and measurable. Since $G_0$ is bounded and measurable, both $G_0\varphi:E_0\to\mathbb{R}$ and $G_0:E_0\to\mathbb{R}$ are bounded [measurable functions](/page/Measurable%20Functions). Therefore
\begin{align*}
\xi_{0,N}(G_0\varphi)\xrightarrow{\mathbb{P}}\xi_0(G_0\varphi)
\end{align*}
and
\begin{align*}
\xi_{0,N}(G_0)\xrightarrow{\mathbb{P}}\xi_0(G_0).
\end{align*}
By hypothesis $\xi_0(G_0)>0$. Applying the ratio result with $X_N=\xi_{0,N}(G_0\varphi)$, $Y_N=\xi_{0,N}(G_0)$, $x=\xi_0(G_0\varphi)$, and $y=\xi_0(G_0)$ gives
\begin{align*}
\eta_{0,N}(\varphi)\xrightarrow{\mathbb{P}}\frac{\xi_0(G_0\varphi)}{\xi_0(G_0)}=\eta_0(\varphi).
\end{align*}
Thus both asserted convergences hold at time $0$.
[/step]
[step:Use the resampling-and-mutation assumption to propagate pre-weighting consistency]
Assume that for some $s\in\{1,\dots,t\}$ and every bounded measurable $\psi:E_{s-1}\to\mathbb{R}$,
\begin{align*}
\eta_{s-1,N}(\psi)\xrightarrow{\mathbb{P}}\eta_{s-1}(\psi).
\end{align*}
This is precisely the hypothesis required by the stated resampling-and-mutation consistency property at time $s$. Hence, for every bounded measurable $\varphi:E_s\to\mathbb{R}$,
\begin{align*}
\xi_{s,N}(\varphi)\xrightarrow{\mathbb{P}}(\eta_{s-1}M_s)(\varphi).
\end{align*}
By the deterministic definition $\xi_s=\eta_{s-1}M_s$, the right-hand side is $\xi_s(\varphi)$. Therefore
\begin{align*}
\xi_{s,N}(\varphi)\xrightarrow{\mathbb{P}}\xi_s(\varphi).
\end{align*}
[/step]
[step:Normalize the predicted empirical measure to obtain post-weighting consistency]
Let $\varphi:E_s\to\mathbb{R}$ be bounded and measurable. Since $G_s:E_s\to[0,\infty)$ is bounded and measurable, the functions $G_s\varphi:E_s\to\mathbb{R}$ and $G_s:E_s\to\mathbb{R}$ are bounded and measurable. From the pre-weighting consistency just obtained,
\begin{align*}
\xi_{s,N}(G_s\varphi)\xrightarrow{\mathbb{P}}\xi_s(G_s\varphi)
\end{align*}
and
\begin{align*}
\xi_{s,N}(G_s)\xrightarrow{\mathbb{P}}\xi_s(G_s).
\end{align*}
The deterministic normalizing constant satisfies $\xi_s(G_s)>0$ by hypothesis. Applying the ratio result with $X_N=\xi_{s,N}(G_s\varphi)$, $Y_N=\xi_{s,N}(G_s)$, $x=\xi_s(G_s\varphi)$, and $y=\xi_s(G_s)$ yields
\begin{align*}
\eta_{s,N}(\varphi)\xrightarrow{\mathbb{P}}\frac{\xi_s(G_s\varphi)}{\xi_s(G_s)}=\eta_s(\varphi).
\end{align*}
This proves the post-weighting consistency at time $s$.
[/step]
[step:Close the induction and evaluate the terminal test function]
The base case at $s=0$ has been proved. The previous two steps show that, whenever post-weighting consistency holds at time $s-1$, pre-weighting consistency and then post-weighting consistency hold at time $s$. By induction, for every $s\in\{0,\dots,t\}$ and every bounded measurable $\varphi:E_s\to\mathbb{R}$,
\begin{align*}
\eta_{s,N}(\varphi)\xrightarrow{\mathbb{P}}\eta_s(\varphi).
\end{align*}
Taking $s=t$ and $\varphi=h$, where $h:E_t\to\mathbb{R}$ is the bounded measurable test function in the statement, gives
\begin{align*}
\eta_{t,N}(h)\xrightarrow{\mathbb{P}}\eta_t(h).
\end{align*}
This is the desired consistency of the terminal sequential Monte Carlo estimate.
[/step]
