On Approximating Probabilities for Small and Large Deviations in $\mathbb{R}^d$

Abstract

A unified approach to approximations of probabilities for sums of $n$ independent random vectors in $\mathbb{R}^d$ is presented based on the Edgeworth expansion of exponentially shifted vectors together with explicit bounds on the errors. Weak conditions are given under which the error bounds may be written as simple order terms in $n$. These results are used in particular to examine approximations to conditional probabilities giving a general method of approximation for these. A number of important special cases are discussed and examined numerically.