A limit formula and recursive algorithm for multivariate Normal tail probability

Abstract

This work develops a formula for the large threshold limit of multivariate Normal tail probability when at least one of the normalised thresholds grows indefinitely. Derived using integration by parts, the formula expresses the tail probability in terms of conditional probabilities involving one less variate, thereby reducing the problem dimension by 1. The formula is asymptotic to Ruben’s formula under Salvage’s condition. It satisfies Plackett’s identity exactly or approximately, depending on the correlation parameter being differentiated. A recursive algorithm is proposed that allows the tail probability limit to be calculated in terms of univariate Normal probabilities only. The algorithm shows promise in numerical examples to offer a semi-analytical approximation under non-asymptotic situations to within an order of magnitude. The number of univariate Normal probability evaluations is at least n!, however, and in this sense the algorithm suffers from the curse of dimension.