Saddlepoint Approximations of Cumulative Distribution Functions of Sums of Random Vectors

In this paper, a real-valued function that approximates the cumulative distribution function (CDF) of a finite sum of real-valued independent and identically distributed random vectors is presented. The approximation error is upper bounded by an expression that is easy to calculate. As a byproduct, an upper bound and a lower bound on the CDF are obtained. Finally, in the case of lattice and absolutely continuous random variables, the proposed approximation is shown to be identical to the saddlepoint approximation of the CDF.