Multinomial and residual offspring resampling

Input a particle count $N\ge 1$, nonnegative finite weights $a_1,\dots ,a_N$, and a scheme label, either multinomial or residual. Let $A={\sum }_{j=1}^N{a}_j$. If $A>0$, set $p_j=a_j\mathrm{/}A$; if $A=0$, set $p_j=1\mathrm{/}N$ for every $j$. Thus $p$ is always a probability vector.

• Multinomial scheme. Draw an index vector $(J_1,\dots ,J_N)\in \{1,\dots ,N{\}}^N$ with conditional mass $\mathbb{P}(J_1=j_1,\dots ,J_N=j_N\mid a)={\prod }_{\mathrm{\ell }=1}^N{p}_{j_{\mathrm{\ell }}}$. Put $C_j=\mathrm{\#}\{\mathrm{\ell }:J_{\mathrm{\ell }}=j\}$.
• Residual scheme. Put $q_j=\lfloor N{p}_j\rfloor$, $r_j=N{p}_j-q_j$, and $R={\sum }_{j=1}^N{r}_j=N-{\sum }_{j=1}^N{q}_j$, an integer in $\{0,\dots ,N-1\}$. If $R=0$, put $C_j=q_j$. If $R>0$, draw $(K_1,\dots ,K_R)$ with conditional mass $\mathbb{P}(K_1=k_1,\dots ,K_R=k_R\mid a)={\prod }_{\mathrm{\ell }=1}^R(r_{k_{\mathrm{\ell }}}\mathrm{/}R)$ and put $C_j=q_j+\mathrm{\#}\{\mathrm{\ell }:K_{\mathrm{\ell }}=j\}$.

For multinomial resampling, return the sampled vector $(J_1,\dots ,J_N)$ in its sampled order, together with its count vector $(C_1,\dots ,C_N)$; do not sort the sampled indices. For residual resampling, return the ancestor list containing exactly $C_j$ copies of index $j$, ordered by increasing index, together with its count vector. Each output is a length-$N$ list and a count vector summing to $N$. When all input weights vanish, the uniform fallback defines the procedure; any downstream normalizer already multiplied by this zero average remains zero.

The multinomial branch is the conditionally IID ancestor-index scheme. The residual branch is a distinct selection law: its deterministic floor counts generally make the child indices dependent, while its conditional expected counts agree with the same probability vector.