Abstract
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 almostsure 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 language  English 

Pages (fromto)  22362260 
Number of pages  25 
Journal  Annals of Statistics 
Volume  47 
Issue number  4 
Early online date  21 May 2019 
DOIs  
Publication status  Published  May 2019 
Keywords
 negative association
 resampling
 Particle filtering
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Dr Mathieu Gerber
 School of Mathematics  Senior Lecturer in Statistical Science
 Statistical Science
Person: Academic