Convergence of sequential quasi-Monte Carlo smoothing algorithms

Gerber and Chopin [J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 (2015) 509-579] recently introduced Sequential quasi-Monte Carlo (SQMC) algorithms as an efficient Way to perform filtering in state-space models. The basic idea is to replace random variables with low-discrepancy point sets, so as to obtain faster convergence than with standard particle filtering. Gerber and Chopin (2015) describe briefly several ways to extend SQMC to smoothing, but do not provide supporting theory for this extension. We discuss more thoroughly how smoothing may be performed within SQMC, and derive convergence results for the so-obtained smoothing algorithms. We consider in particular SQMC equivalents of forward smoothing and forward filtering backward sampling, which are the most well-known smoothing techniques. As a preliminary step, we provide a generalization of the classical result of Hlawka and nick [Computing (Arch. Elektron. Rechnen) 9 (1972) 127-138] on the transformation of QMC point sets into low discrepancy point sets with respect to non uniform distributions. As a corollary of the latter, we note that we can slightly weaken the assumptions to prove the consistency of SQMC.