Full-resampling fixed-level SIR splitting with invariant mutations

Input the nested levels of Nested rare-event levels and conditional laws on a standard Borel state space $(\mathrm{\Omega },\mathcal{F})$, a particle count $N\ge 1$, and probability kernels $K_{\mathrm{\ell }}$ for $1\le \mathrm{\ell }<L$. Each mutation kernel must satisfy $K_{\mathrm{\ell }}(x,A_{\mathrm{\ell }})=1(x\in A_{\mathrm{\ell }}),{\pi }_{\mathrm{\ell }}{K}_{\mathrm{\ell }}={\pi }_{\mathrm{\ell }}\mathrm{.}$ If a required kernel is unavailable or violates its declared conditions, return mutation-kernel-unavailable.

Apply the canonical full-resampling sequential importance-resampling procedure of Sequential importance resampling for Feynman–Kac path targets to the following finite Feynman–Kac model: initial law $P$, initial potential $G_0\equiv 1$, identity transition $M_1$, event potential $G_1={\mathbf{1}}_{A_1}$, and, for $2\le \mathrm{\ell }\le L$, transition $M_{\mathrm{\ell }}=K_{\mathrm{\ell }-1}$ and potential $G_{\mathrm{\ell }}={\mathbf{1}}_{A_{\mathrm{\ell }}}$. Use multinomial full resampling at every selection and proposal kernels equal to the target transitions, so each Radon–Nikodym correction is one.

Return the SIR normalizer estimate $\widehat{p}$ and the full particle output. If any incremental average event weight is zero, report status zero-particle-prefix and $\widehat{p}=0$, as in the canonical SIR procedure. This status does not assert that the target probability is zero. The exact conditional level sampler is not inferred from the existence of an invariant kernel; kernel accessibility is an input requirement.