The Bayesian estimation of the unknown parameters of state-space (dynamical)systems has received considerable attention over the past decade, with ahandful of powerful algorithms being introduced. In this paper we tackle thetheoretical analysis of the recently proposed {\it nonlinear} population MonteCarlo (NPMC). This is an iterative importance sampling scheme whose keyfeatures, compared to conventional importance samplers, are (i) the approximatecomputation of the importance weights (IWs) assigned to the Monte Carlo samplesand (ii) the nonlinear transformation of these IWs in order to prevent thedegeneracy problem that flaws the performance of conventional importancesamplers. The contribution of the present paper is a rigorous proof ofconvergence of the nonlinear IS (NIS) scheme as the number of Monte Carlosamples, $M$, increases. Our analysis reveals that the NIS approximation errorsconverge to 0 almost surely and with the optimal Monte Carlo rate of$M^{-\frac{1}{2}}$. Moreover, we prove that this is achieved even when the meanestimation error of the IWs remains constant, a property that has been termed{\it exact approximation} in the Markov chain Monte Carlo literature. Weillustrate these theoretical results by means of a computer simulation exampleinvolving the estimation of the parameters of a state-space model typicallyused for target tracking.
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