  In this work, we prove the joint convergence in distribution of $q$ variables
modulo one obtained as partial sums of a sequence of i.i.d. square integrable
random variables multiplied by a common factor given by some function of an
empirical mean of the same sequence. The limit is uniformy distributed over
$[0,1]^q$. To deal with the coupling introduced by the common factor, we assume
that the joint distribution of the random variables has a non zero component
absolutely continuous with respect to the Lebesgue measure, so that the
convergence in the central limit theorem for this sequence holds in total
variation distance. While our result provides a generalization of Benford's law
to a data adapted mantissa, our main motivation is the derivation of a central
limit theorem for the stratified resampling mechanism, which is performed in
the companion paper \cite{echant}.
