The sequential particle estimator is unbiased for Feynman–Kac path integrals

First, extend each setwise density identity needed here to nonnegative measurable integrands. If $P(A)={\int }_{A}udQ$ for every measurable $A$, with $u\ge 0$, choose simple $g_n\uparrow g$ by [F15]. The simple-function identity [F14] gives $\int g_{n}dP=\int g_{n}udQ$ for each $n$, and [F16] on both sides yields $\int gdP=\int gudQ$, allowing infinity. This applies to the initial law pair and, for almost every preceding proposal state, to each transition-kernel pair in [F5]. At time zero, the algorithm gives ${\widehat{\gamma }}_0^N(\phi )=N^{-1}{\sum }_{i=1}^N{G}_0(X_0^i){\rho }_0(X_0^i)\phi (X_0^i)$, with zero-weight terms interpreted by the nonnegative-integral convention $0\cdot \mathrm{\infty }=0$. Each $X_0^i$ has law ${\nu }_0$, so [F6], finite-sum linearity [F7], and the just-proved extension give $\mathbb{E}[{\widehat{\gamma }}_0^N(\phi )]=\int G_0{\rho }_0\phi d{\nu }_0={\gamma }_0(\phi )$. The finite target integral makes this random variable integrable.

The finite procedure defines measurable transition kernels. The initial IID draws from ${\nu }_0$ form a finite product kernel by [F21]. At each later step, the previous normalized weights are measurable functions of the previous paths by [F17, F18, F19, F20]. The multinomial and residual laws are probability kernels on the finite ancestor-index set: normalization uses division on the Borel branch where the sum is positive, each floor map is Borel since $\{x:\lfloor x\rfloor =m\}=[m,m+1)$, and the $R=0$ or $R>0$ branches are Borel. Since $R$ takes only the values $0,\dots ,N-1$, each residual output-list probability is a finite sum over possible draw vectors of Borel product masses. For a fixed ancestor vector, the proposal law of the next particles is the finite product of the pullbacks of the $R_k$ kernels along measurable coordinate projections; [F17, F18, F19] make each pullback a probability kernel, and [F21] makes their product a probability kernel. The finite mixture over ancestor vectors is again a probability kernel by finite sums and products [F20]. Thus [F12] applies to the particle-system transitions.

Every time-zero particle path has law $Q_0$. Inductively, suppose each previous particle path avoids every fixed $Q_{k-1}$-null history set almost surely. If $Q_k(A)=0$, the pullback of $R_k$ to the history space is a probability kernel by step 1.2, and [F11] makes $x_{0:k-1}\mapsto R_k(x_{k-1},A_{x_{0:k-1}})$ measurable. The proposal path recursion [F12] and zero-integral criterion [F13] show it vanishes for $Q_{k-1}$-almost every history. The kernel-derivative exceptional parent-state set has a null preimage under the last-coordinate marginal of $Q_{k-1}$, so every stored path avoids that preimage; resampling copies only existing paths, and propagation therefore avoids $A$. Each initial or kernel derivative has reference integral one by [F5]; it is therefore finite almost everywhere for its reference law. The finite-valued potentials and the finite particle count then make all particle weights finite almost surely.

Write ${\mathcal{F}}_{k-1}$ for the previous particle histories and weights. Let $H_k$ be the integral of ${\widehat{\gamma }}_k^N(\phi )$ against the algorithm's conditional finite-step transition law. By [F3], the expected number of children from parent $j$ is $N{\bar{w}}_{k-1}^j$. The algorithm gives ${\widehat{Z}}_k{\bar{w}}_k^i={\widehat{Z}}_{k-1}{a}_k^i\mathrm{/}N$ when the new weight sum is positive; when it is zero, both sides are zero. For each parent outside the exceptional proposal-marginal null set, [F18, F19, F20] make $x_k\mapsto G_k(x_k)\phi (x_{0:k})$ nonnegative measurable; apply the simple-approximation extension in step 1.1 to it. The setwise identity in [F5] then gives $\int G_k\phi r_{k}d{R}_k=\int G_k\phi d{M}_k$, including infinite values. Integrating each child's incremental weight and summing the expected counts therefore gives $H_k={\widehat{\gamma }}_{k-1}^N(g_k)$, where $g_k(x_{0:k-1})=\int G_k(x_k)\phi (x_{0:k})M_k(x_{k-1},d{x}_k)$. If the previous normalizer estimate is zero, both sides vanish by the algorithm's zero-weight convention. Measurability follows from [F11]. This is initially a nonnegative conditional-kernel integral, not yet an $L^1$ conditional expectation.

By [F4], the target integral of $g_k$ is ${\gamma }_{k-1}(g_k)={\gamma }_k(\phi )<\mathrm{\infty }$. The induction hypothesis therefore gives $\mathbb{E}[H_k]={\gamma }_k(\phi )$. For every prior event $B\in {\mathcal{F}}_{k-1}$, the ordered joint law of the prior histories and the algorithm's conditional finite-step transition law [F12], integrated with the nonnegative integrand ${\mathbf{1}}_B{\widehat{\gamma }}_k^N(\phi )$, gives $\mathbb{E}[{\mathbf{1}}_B{\widehat{\gamma }}_k^N(\phi )]=\mathbb{E}[{\mathbf{1}}_B{H}_k]\mathrm{.}$ This identity is valid before any integrability assumption, by nonnegative iterated integration. Taking $B=\mathrm{\Omega }$ shows that ${\widehat{\gamma }}_k^N(\phi )$ has finite expectation ${\gamma }_k(\phi )$, so it is in $L^1$; $H_k$ is also in $L^1$ by its finite expectation. Thus the event identities and measurability make $H_k$ a version of $\mathbb{E}[{\widehat{\gamma }}_k^N(\phi )\mid {\mathcal{F}}_{k-1}]$ by [F2] and [F10]. The tower property [F9] now yields $\mathbb{E}[{\widehat{\gamma }}_k^N(\phi )]={\gamma }_k(\phi )$, completing the induction step.

Apply the established identity at $k=t$ to the constant function $\phi =1$. Then ${\widehat{\gamma }}_t^N(1)={\widehat{Z}}_t{\widehat{\pi }}_t^N(1)={\widehat{Z}}_t$, while ${\gamma }_t(1)=Z_t$, proving the stated normalizer unbiasedness.