  This paper is devoted to establishing exponential bounds for the
probabilities of deviation of a sample sum from its expectation, when the
variables involved in the summation are obtained by sampling in a finite
population according to a rejective scheme, generalizing sampling without
replacement, and by using an appropriate normalization. In contrast to Poisson
sampling, classical deviation inequalities in the i.i.d. setting do not
straightforwardly apply to sample sums related to rejective schemes, due to the
inherent dependence structure of the sampled points. We show here how to
overcome this difficulty, by combining the formulation of rejective sampling as
Poisson sampling conditioned upon the sample size with the Escher
transformation. In particular, the Bennett/Bernstein type bounds established
highlight the effect of the asymptotic variance of the (properly standardized)
sample weighted sum and are shown to be much more accurate than those based on
the negative association property shared by the terms involved in the
summation. Beyond its interest in itself, such a result for rejective sampling
is crucial, insofar as it can be extended to many other sampling schemes,
namely those that can be accurately approximated by rejective plans in the
sense of the total variation distance.
