Negative association, ordering and convergence of resampling methods

Mathieu Gerber, Nicolas Chopin, Nick Whiteley

Research output: Contribution to journalArticle (Academic Journal)peer-review

32 Citations (Scopus)
311 Downloads (Pure)


We study convergence and convergence rates for resampling schems. Our first main result is a general consistency theorem based on the notion of negative association, which is applied to establish the almost-sure weak convergence of measures output from Kitagawa's (1996) stratified resampling method. Carpenter et al's (1999) systematic resampling method is similar in structure but can fail to converge depending on the order of the input samples. We introduce a new resampling algorithm based on a stochastic rounding technique of Srinivasan (2001), which shares some attractive properties of systematic resampling, but which exhibits negative association and therefore converges irrespective of the order of the input samples. We confirm a conjecture made by Kitagawa (1996) that ordering input samples by their states in $\mathbb{R}$ yields a faster rate of convergence; we establish that when particles are ordered using the Hilbert curve in $\mathbb{R}^d$, the variance of the resampling error is ${\scriptscriptstyle\mathcal{O}}(N^{-(1+1/d)})$ under mild conditions, where $N$ is the number of particles. We use these results to establish asymptotic properties of particle algorithms based on resampling schemes that differ from multinomial resampling.
Original languageEnglish
Pages (from-to)2236-2260
Number of pages25
JournalAnnals of Statistics
Issue number4
Early online date21 May 2019
Publication statusPublished - May 2019


  • negative association
  • resampling
  • Particle filtering


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