We provide a comprehensive characterisation of the theoretical properties of the divide-and-conquer sequential Monte Carlo (DaC-SMC) algorithm. We firmly establish it as a well-founded method by showing that it possesses the same basic properties as conventional sequential Monte Carlo (SMC) algorithms do. In particular, we derive pertinent laws of large numbers, $L^p$ inequalities, and central limit theorems; and we characterize the bias in the normalized estimates produced by the algorithm and argue the absence thereof in the unnormalized ones. We further consider its practical implementation and several interesting variants; obtain expressions for its globally and locally optimal intermediate targets, auxiliary measures, and proposal kernels; and show that, in comparable conditions, DaC-SMC proves more statistically efficient than its direct SMC analogue. We close the paper with a discussion of our results, open questions, and future research directions.

翻译：本文全面刻画了分治序贯蒙特卡罗（DaC-SMC）算法的理论性质。我们通过证明该算法具备与传统序贯蒙特卡罗（SMC）算法相同的基本性质，将其牢固确立为一种具有坚实理论基础的方法。具体而言，我们推导了相应的大数定律、$L^p$不等式以及中心极限定理；同时表征了算法产生的归一化估计中的偏差，并论证了非归一化估计中不存在偏差。我们进一步讨论了算法的实际实现及若干有趣的变体；获得了其全局与局部最优中间目标、辅助测度以及提议核的表达式；并证明在可比条件下，DaC-SMC比其直接SMC模拟具有更高的统计效率。文章最后讨论了研究结果、开放性问题及未来研究方向。