On Large Deviations in the Poisson Approximation

Abstract

This paper proves a general lemma comparing the behavior of probabilities of large deviations ${\bf P}(X \geqq x)$ 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 kth order: \[ \left| {\widetilde\Gamma _k (X)} \right| \leqq \frac{{k!\lambda }}{{\Delta ^{k - 1} }}\quad \text{for } \forall k \geqq 2 \] for some $\Delta > 0$, then large deviations may be compared in the interval $1 \leqq x - \lambda < \delta \lambda \Delta , 0 < \delta < 1$. For such x\[ \frac{{{\bf P} ( {X \geqq x} )}} {{1 - P ( {x,\lambda } )}} = e^{L( x )} \left( 1 + \theta _1 \frac{{1 + \lambda }} {x} + \theta _2 \frac{{( {x - \lambda } )^{3/ 2}}}{\Delta } \right), \] where $L(x)$ is a power series and $| {\theta _i } | < C(\delta ), i = 1,2$.