Laplace-Gauss Integrals, Gaussian Measure Asymptotic Behaviour and Probabilities of Moderate Deviations

Abstract

There is given a general Gaussian measure representation on arbitrary finite dimensional Borel sets. This representation reflects B. Cavalieri’s and E. Torricelli’s "indivisible method" in a modern language. Based upon it, assertions are derived about the Gaussian measure asymptotic behaviour on Borel sets whose distance from the origin tends to infinity. Also two specific multivariate moderate deviation limit theorems for sums of i.i.d. random vectors are deduced.