Fixed-lag particle smoother

For a finite horizon $T\ge 0$, fix an integer lag $L\ge 0$, a path count $S\ge 1$, and the normalized filtering particle clouds and transition inputs required by Particle backward simulator. For each time $t\in \{0,\dots ,T\}$, put $s_t=\min (t+L,T)$. Independently for each $t$, run the particle backward simulator using only the filtering clouds at times $0,\dots ,s_t$ and request $S$ paths. Retain their time-$t$ coordinates ${\tilde{X}}_t^{t,1},\dots ,{\tilde{X}}_t^{t,S}$ and return the empirical measure ${\widehat{\eta }}_{t\mathrm{\mid }{s}_t}^S=\frac{1}{S}{\sum }_{r=1}^S{\delta }_{{\tilde{X}}_t^{t,r}}\mathrm{.}$ In particle mode this empirical law approximates $\mathcal{L}(X_t\mid Y_{1:s_t}),$ not the full smoothing law $\mathcal{L}(X_t\mid Y_{1:T})$ unless $s_t=T$ or the two conditional laws happen to agree. The same procedure can use supplied exact backward kernels when the model has no transition density; for each $s_t$ it uses the supplied filtering kernel at $y_{1:s_t}$ and the backward kernels for the corresponding prefix histories. On histories where these kernels factor the truncated smoother, it samples that exact conditional law.

The procedure returns one truncated smoothing marginal per $t$. It does not assert a bound on the lag bias, a forgetting rate, or convergence to the full-data smoother for fixed $L$. Those conclusions require separate conditions on the model.