We study the empirical measure associated to a sample of size $n$ and modified by $N$ iterations of the raking-ratio method. This 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. We study nonasymptotic properties by using a Gaussian approximation which yields uniform Berry-Esseen type bounds depending on $n, N$ and provides estimates of the uniform quadratic risk reduction. A closed-form expression of the limiting covariance matrices is derived as $N\rightarrow +\infty$. In the two-way contingency table case the limiting process has a simple explicit formula.

46 pages