Self-normalized importance sampling is strongly consistent under finite first moments

Use [F8] with density $r=q$ for integration against $Q$ and with $r=\tilde{p}\mathrm{/}{Z}_p$ for integration against $\pi$. By [F6], the functions $w(x)=\tilde{p}(x)\mathrm{/}q(x)$ on $q>0$ (with value $0$ on $q=0$) and $v(x)=w(x)f(x)$ are measurable. The support condition gives $\pi (\{q=0\})=0$. For each $i$, $X_i$ has law $Q$, so the given finite first moment applies to its weight and integrand. By [F2], ${\mathbb{E}}_Q[W_i\mathrm{\mid }f(X_i)\mathrm{\mid }]={\int }_{\{q>0\}}\frac{\tilde{p}(x)\mathrm{\mid }f(x)\mathrm{\mid }}{q(x)}q(x)d\mu (x)={\int }_{\{q>0\}}\tilde{p}\mathrm{\mid }f\mathrm{\mid }d\mu \mathrm{.}$ The given finite moment therefore implies ${\int }_{\{q>0\}}\tilde{p}\mathrm{\mid }f\mathrm{\mid }d\mu <\mathrm{\infty }$. Since $\pi (\{q=0\})=0$, [F7] gives ${\int }_{\{q=0\}}\mathrm{\mid }f\mathrm{\mid }d\pi =0$, and thus ${\int }_{\{q=0\}}\tilde{p}\mathrm{\mid }f\mathrm{\mid }d\mu =Z_p{\int }_{\{q=0\}}\mathrm{\mid }f\mathrm{\mid }d\pi =0$. Hence ${\int }_{\mathsf{X}}\tilde{p}\mathrm{\mid }f\mathrm{\mid }d\mu <\mathrm{\infty }$ and $\theta =\int fd\pi$ is well-defined. For any real measurable $h$ whose absolute integral is finite against $rd\mu$, apply [F8] separately to $h^{+}=\max (h,0)$ and $h^{-}=\max (-h,0)$: both product integrals are finite because $h^{\pm }\le \mathrm{\mid }h\mathrm{\mid }$. Subtracting their finite identities yields $\int hd(rd\mu )=\int hrd\mu$, since $(hr{)}^{\pm }=h^{\pm }r$ for $r\ge 0$ (with the density convention $0\cdot \mathrm{\infty }=0$). The integrable case of [F2] now gives ${\mathbb{E}}_Q{V}_i={\int }_{\{q>0\}}\frac{\tilde{p}(x)f(x)}{q(x)}q(x)d\mu (x)={\int }_{\{q>0\}}\tilde{p}fd\mu ={\int }_{\mathsf{X}}\tilde{p}fd\mu =Z_p\theta ,$ where the omitted signed integral is zero because its absolute integral over $\{q=0\}$ is zero. Likewise, the nonnegative case of [F2] gives ${\mathbb{E}}_Q{W}_i={\int }_{\{q>0\}}\tilde{p}d\mu =Z_p$, since ${\int }_{\{q=0\}}\tilde{p}d\mu =Z_p\pi (\{q=0\})=0$. All these expectations agree across $i$ by the common law of the $X_i$.

Applying [F3] separately to the measurable maps $w$ and $v$ shows that each family $(W_i)$ and $(V_i)$ is independent. Each family has a common law because every $X_i$ has law $Q$ and the same map is applied to each draw. Thus both sequences are IID as in [F4], and [F5] gives $n^{-1}{\sum }_i{W}_i\to Z_p$ and $n^{-1}{\sum }_i{V}_i\to Z_p\theta$ almost surely.

On the probability-one event from step 1.2, the denominator average is eventually greater than $Z_p\mathrm{/}2>0$, so the ratio is eventually defined and ${\widehat{\theta }}_{\mathrm{S}\mathrm{N},n}=\frac{n^{-1}{\sum }_i{V}_i}{n^{-1}{\sum }_i{W}_i}⟶\frac{Z_p\theta }{Z_p}=\theta$. The numerical zero interpretation for any preceding finite prefix with zero denominator is the convention stated in this theorem; the algorithm itself returns the no-estimate status in [F1].