Variance scaling with both M and P in waste-free SMC

Investigate whether waste-free Sequential Monte Carlo estimators can achieve an effective 1/(M P) variance or concentration rate, rather than a 1/P rate, when using M parallel chains of length P despite the dependence induced by resampling.

Background

Waste-free SMC produces N = M P samples per iteration by reweighting all intermediate states of M chains of length P, but the resampling step induces dependence across chains and within chains.

The present finite-sample analysis effectively treats M as a constant and yields bounds scaling in 1/P; it is unknown whether the dependence structure still allows a 1/(M P) rate analogous to IID sampling.

References

The limitation stems from dependence induced by resampling, and it remains unclear whether one can recover an effective $1/(MP)$ variance rate rather than a $1/P$ rate.

— On the complexity of standard and waste-free SMC samplers  (2604.03352 - Fay et al., 3 Apr 2026) in Section 7.2, Future work