Convergence of random-weight sequential Monte Carlo methods

We investigate the properties of a sequential Monte Carlo method where the particle weight that appears in the algorithm is estimated by a positive, unbiased estimator. We present broadly-applicable convergence results, including a central limit theorem, and we discuss their relevance for applications in statistical physics. Using these results, we show that the resampling step reduces the impact of the randomness of the weights on the asymptotic variance of the estimator. In addition, we explore the limits of convergence of the sequential Monte Carlo method, with a focus on almost sure convergence. We construct an example algorithm where we can prove convergence in probability, but which does not converge almost surely, even in the non-random-weight case.