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In this work, we prove the joint convergence in distribution of $q$ variablesmodulo one obtained as partial sums of a sequence of i.i.d. square integrablerandom variables multiplied by a common factor given by some function of anempirical 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 assumethat the joint distribution of the random variables has a non zero componentabsolutely continuous with respect to the Lebesgue measure, so that theconvergence in the central limit theorem for this sequence holds in totalvariation distance. While our result provides a generalization of Benford's lawto a data adapted mantissa, our main motivation is the derivation of a centrallimit theorem for the stratified resampling mechanism, which is performed inthe companion paper \cite{echant}.

Convergence to the uniform distribution of vectors of partial sums modulo one with a common factor