Asymptotic genealogies of interacting particle systems with an application to sequential Monte Carlo
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We consider weighted particle systems of fixed size, in which a new generation of particles is formed by resampling particles from the current generation with probabilities proportional to their weights. This set-up covers a broad class of sequential Monte Carlo (SMC) methods, which are widely-used in applied statistics and cognate disciplines across a range of domains. We consider the genealogical tree embedded into such particle systems by the resampling operation, and identify conditions, as well as an appropriate time-scaling, under which it converges to the Kingman coalescent in the infinite system size limit in the sense of finite-dimensional distributions. This makes the plethora of distributional results known for the Kingman coalescent available for analysis of SMC algorithms, which we illustrate by characterising the limiting mean and variance of the tree height, as well as of the total branch length of the tree. It also greatly improves the tractability of genealogies of SMC methods, which are known to be closely connected to the performance of these algorithms. The conditions which we require to prove convergence are strong, but we demonstrate by simulation that they do not appear to be necessary.
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