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Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. (English) Zbl 1079.65006
A particle system is a collection $$(\vartheta^{(j,H)},w^{(j,H)})_{j\leq H}$$ where $$\vartheta^{(j,H)}\in\Theta$$ are “particles” and $$w^{(j,H)}>0$$ are their “weights”. The system targets a distribution $$\pi$$ on $$\Theta$$ if for any measurable $$\varphi$$ with $$| \mathbf{E}_\pi (\varphi)| <\infty$$, $\hat E_H(\varphi)= { \sum_{j=1}^H w^{(j,H)}\varphi(\vartheta^{(j,H)}) \over \sum_{j=1}^H w^{(j,H)} } \to \mathbf{E}_\pi (\varphi).$ A sequential Monte Carlo algorithm (a particle filter) produces recursively (using mutation-correction-resampling scheme) a sequence of particle systems which target a sequence of distributions $$\pi_t$$ on $$\Theta_t$$. In the Bayes estimation problems $$\Theta_t=\Theta$$ is the parameter space and $$\pi_t$$ is an a posteriori distribution of the parameter $$\vartheta$$ given the sample of size $$t$$. In the state-space filtering or smoothing $$\Theta_t$$ is the space of states trajectories and $$\pi_t$$ is the conditional distribution of the trajectory given the data.
The author obtains conditions for the central limit theorem of the form $$\sqrt{H}(\hat E_H(\varphi)-\mathbf{E}_\pi (\varphi)) \Rightarrow N(0,V_t(\varphi))$$ where $$V_t(\varphi)$$ is described using recursive formulae. These conditions hold for many of sequential Monte Carlo algorithms including the resample-move algorithm and the residual resampling scheme. Asymptotics of $$V_t(\varphi)$$ as $$t\to\infty$$ are investigated for Bayesian problems.

##### MSC:
 65C05 Monte Carlo methods 62F15 Bayesian inference 60F05 Central limit and other weak theorems 62L10 Sequential statistical analysis
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