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In a recent paper Beskos et al (2011), the Sequential Monte Carlo (SMC)sampler introduced in Del Moral et al (2006), Neal (2001) has been shown to beasymptotically stable in the dimension of the state space d at a cost that isonly polynomial in d, when N the number of Monte Carlo samples, is fixed. Moreprecisely, it has been established that the effective sample size (ESS) of theensuing (approximate) sample and the Monte Carlo error of fixed dimensionalmarginals will converge as $d$ grows, with a computational cost of$\mathcal{O}(Nd^2)$. In the present work, further results on SMC methods inhigh dimensions are provided as $d\to\infty$ and with $N$ fixed. We deduce anexplicit bound on the Monte-Carlo error for estimates derived using the SMCsampler and the exact asymptotic relative $\mathbb{L}_2$-error of the estimateof the normalizing constant. We also establish marginal propagation of chaosproperties of the algorithm. The accuracy in high-dimensions of someapproximate SMC-based filtering schemes is also discussed.

Error Bounds and Normalizing Constants for Sequential Monte Carlo in High Dimensions