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This paper is devoted to establishing exponential bounds for theprobabilities of deviation of a sample sum from its expectation, when thevariables involved in the summation are obtained by sampling in a finitepopulation according to a rejective scheme, generalizing sampling withoutreplacement, and by using an appropriate normalization. In contrast to Poissonsampling, classical deviation inequalities in the i.i.d. setting do notstraightforwardly apply to sample sums related to rejective schemes, due to theinherent dependence structure of the sampled points. We show here how toovercome this difficulty, by combining the formulation of rejective sampling asPoisson sampling conditioned upon the sample size with the Eschertransformation. In particular, the Bennett/Bernstein type bounds establishedhighlight the effect of the asymptotic variance of the (properly standardized)sample weighted sum and are shown to be much more accurate than those based onthe negative association property shared by the terms involved in thesummation. Beyond its interest in itself, such a result for rejective samplingis crucial, insofar as it can be extended to many other sampling schemes,namely those that can be accurately approximated by rejective plans in thesense of the total variation distance.

Sharp exponential inequalities in survey sampling: conditional Poisson sampling schemes