## Section: New Results

### Particle methods with adaptive resampling

Participant : François Le Gland.

This is a collaboration with Élise Arnaud, from université Joseph Fourier and INRIA Grenoble Rhône Alpes (project–team PERCEPTION).

A longstanding problem in particle or sequential Monte Carlo (SMC) methods is to mathematically prove the popular belief that resampling does improve the performance of the estimation (this of course is not always true, and the real question is to clarify classes of problems where resampling helps). A more pragmatic answer to the problem is to use adaptive procedures that have been proposed on the basis of heuristic considerations, where resampling is performed only when it is felt necessary, i.e. when some criterion (effective number of particles, entropy of the sample, etc.) reaches some prescribed threshold. It still remains to mathematically prove the efficiency of such adaptive procedures. Our first contribution has been to consider a design where resampling is performed at some intermediate fixed time instants, and to optimize the asymptotic variance of the estimation error w.r.t. the resampling time instants. The second contribution [15] has been to prove a central limit theorem for particle methods with adaptive resampling, using an interpretation of particle methods where importance weights are interpreted as particles [62] , as long as they are not used for resampling purpose, and to minimize the asymptotic variance w.r.t. the threshold.