The absolutely continuous assumption on the Markov transitions of
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are only used to derive in an informal (and rather abusive) way different formulae between posterior distributions using the Bayes’ rule for conditional densities. Since resampling is not dependent on any particular application, the A-SIR analysis is appropriate for any type of particle filtering algorithm that adopts a resampling procedure. People only needed a short term time series of interest for statistical analysis \[[@pone. Unable to display preview. The authors named their algorithm ‘the bootstrap filter’, and demonstrated that compared to other filtering methods, their bootstrap algorithm does not require any assumption about that state-space or the noise of the system. $\left.
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Protter. The first uniform convergence results with respect to the time parameter for particle filters were developed in the end of the 1990s by Pierre Del Moral and Alice Guionnet. For instance, the evolution of the one-step optimal predictor
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{\displaystyle \eta _{n}(dx_{n})=p(x_{n}|y_{0},\cdots ,y_{n-1})dx_{n}}
satisfies a nonlinear evolution starting with the probability distribution
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