Large deviations for subsampling from individual sequences

Consider a sequence of m deterministic points in ##R##d, and consider the empirical measure of a random sample (without replacements) of size n = n(m). We prove the large deviation principle and compute the resulting rate function for the latter empirical measure under the assumptions that the empirical measure of the m-sequence converges and that n/m tends to some 0 < [beta] < 1. Surprisingly, the resulting rate function differs from the one in Sanov's theorem for the large deviations of the empirical measure of an i.i.d. sample. The proof uses coupling and allows for considering many other sampling schemes as well as deriving the corresponding moderate deviation results.