Auxiliary information : the raking-ratio empirical process

We study the empirical measure associated to a sample of size $n$ and modified by $N$ iterations of the raking-ratio method. The empirical measure is adjusted to match the true probability of sets in a finite partition which changes each step. We establish asymptotic properties of the raking-ratio empirical process indexed by functions as $n\rightarrow +\infty$, for $N$ fixed. A closed-form expression of the limiting covariance matrices is derived as $N\rightarrow +\infty$. The nonasymptotic Gaussian approximation we use also yields uniform Berry-Esseen type bounds in $n, N$ and sharp estimates of the uniform quadratic risk reduction. In the two-way contingency table formulas characterizing the limiting process are very simple.