  In the last decade, sequential Monte-Carlo methods (SMC) emerged as a key
tool in computational statistics. These algorithms approximate a sequence of
distributions by a sequence of weighted empirical measures associated to a
weighted population of particles. These particles and weights are generated
recursively according to elementary transformations: mutation and selection.
Examples of applications include the sequential Monte-Carlo techniques to solve
optimal non-linear filtering problems in state-space models, molecular
simulation, genetic optimization, etc.
  Despite many theoretical advances the asymptotic property of these
approximations remains of course a question of central interest. In this paper,
we analyze sequential Monte Carlo methods from an asymptotic perspective, that
is, we establish law of large numbers and invariance principle as the number of
particles gets large. We introduce the concepts of "weighted sample"
consistency and asymptotic normality, and derive conditions under which the
mutation and the selection procedure used in the sequential Monte-Carlo
build-up preserve these properties. To illustrate our findings, we analyze SMC
algorithms to approximate the filtering distribution in state-space models. We
show how our techniques allow to relax restrictive technical conditions used in
previously reported works and provide grounds to analyze more sophisticated
sequential sampling strategies.
