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We provide a comprehensive characterisation of the theoretical properties ofthe divide-and-conquer sequential Monte Carlo (DaC-SMC) algorithm. We firmlyestablish it as a well-founded method by showing that it possesses the samebasic properties as conventional sequential Monte Carlo (SMC) algorithms do. Inparticular, we derive pertinent laws of large numbers, $L^p$ inequalities, andcentral limit theorems; and we characterize the bias in the normalizedestimates produced by the algorithm and argue the absence thereof in theunnormalized ones. We further consider its practical implementation and severalinteresting variants; obtain expressions for its globally and locally optimalintermediate targets, auxiliary measures, and proposal kernels; and show that,in comparable conditions, DaC-SMC proves more statistically efficient than itsdirect SMC analogue. We close the paper with a discussion of our results, openquestions, and future research directions.

The divide-and-conquer sequential Monte Carlo algorithm: theoretical properties and limit theorems