Convergence to the uniform distribution of vectors of partial sums modulo one with a common factor
Abstract
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 [15].
Domains
Mathematics [math]Origin | Files produced by the author(s) |
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