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On large deviations in the Poisson approximation. (English. Russian original) Zbl 0807.60035

Theory Probab. Appl. 38, No. 2, 385-393 (1993); translation from Teor. Veroyatn. Primen. 38, No. 2, 460-470 (1993).
Summary: This paper proves a general lemma comparing the behavior of probabilities of large deviations \({\mathbf P}(X \geq x)\) of a random variable \(X\) against the Poisson distribution \(1 - P(x,\lambda)\) (\(\lambda\) is the parameter of the Poisson distribution). When upper bounds are known for the factorial cumulants \(\widetilde{\Gamma}_ k (x)\) of \(k\)th order \[ | \widetilde{\Gamma}_ k(X)| \leq {k! \lambda \over \Delta^{k - 1}}\qquad \forall k \geq 2 \] for some \(\Delta > 0\), then large deviations may be compared in the interval \(1 \leq x - \lambda < \delta \lambda \Delta\), \(0 < \delta < 1\). For such \(x\), \[ {{\mathbf P}(X \geq x)\over 1 - P(x,\lambda)} = e^{L(x)} \left(1 + \theta_ 1 {1 + \lambda \over x} + \theta_ 2 {(x - \lambda)^{3/2} \over \Delta}\right), \] where \(L(x)\) is a power series and \(| \theta_ i| < C(\delta)\), \(i = 1,2\).

MSC:

60F10 Large deviations
60G50 Sums of independent random variables; random walks
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