  The stratified resampling mechanism is one of the resampling schemes commonly
used in the resampling steps of particle filters. In the present paper, we
prove a central limit theorem for this mechanism under the assumption that the
initial positions are independent and identically distributed and the weights
proportional to a positive function of the positions such that the image of
their common distribution by this function has a non zero component absolutely
continuous with respect to the Lebesgue measure. This result relies on the
convergence in distribution of the fractional part of partial sums of the
normalized weights to some random variable uniformly distributed on $[0,1]$,
which is established in the companion paper \cite{CLTfract} by overcoming the
difficulty raised by the coupling through the normalization. Under the
conjecture that a similar convergence in distribution remains valid at the next
steps of a particle filter which alternates selections according to the
stratified resampling mechanism and mutations according to Markov kernels, we
provide an inductive formula for the asymptotic variance of the resampled
population after $n$ steps. We perform numerical experiments which support the
validity of this formula.
