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A bound on the moment generating function of a sum of dependent variables with an application to simple random sampling without replacement. (English) Zbl 0796.60020
Summary: We prove the following inequality: Let \(\{x_ i\}\) be an arbitrary sequence of random variables. Then there exists a \(\sigma\)-field \({\mathcal G}\), and a \({\mathcal G}\)-conditionally independent sequence \(\{y_ i\}\) tangent to \(\{x_ i\}\) (in particular, \(y_ i\) has the same distribution as \(x_ i\) for all \(i\)) such that for all \(\lambda\) \[ E\exp\Bigl(\lambda\sum^ n_{i=1}x_ i\Bigr)\leqq\sqrt{E\exp\Bigl(2\lambda\sum^ n_{ i=1}y_ i\Bigr)}.\tag{*} \] As application of the above we show that the tail behaviour of \(\sum^ n_{i=1}y_ i\) controls the tail behaviour of \(\sum^ n_{i=1}x_ i\) whenever the conditionally independent sequence is sub- Gaussian. Furthermore, by considering \(\lambda\) as a random variable independent of \(\{x_ i\}\), \(\{y_ i\}\) we show that (*) implies several new decoupling inequalities including a new result for \(l_ 2\) valued random variables. Making the theory of decoupling available to the mainstream of statistics we give several examples where the conditionally independent sequence can be identified, and introduce conditionally independent sampling as an alternative sampling scheme to sampling without replacement from finite populations.

MSC:
60E15 Inequalities; stochastic orderings
60G42 Martingales with discrete parameter
60G40 Stopping times; optimal stopping problems; gambling theory
60G50 Sums of independent random variables; random walks
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