Integral representations for the joint survival functions of the cumulated components of multinomial random vectors

TL;DR

This paper derives a multivariate normal integral expression for the joint survival function of cumulated components of multinomial vectors, linking it to Dirichlet distributions, with two different proof methods provided.

Contribution

It introduces a novel integral representation for the joint survival function of multinomial components, extending previous inequalities and establishing a connection with Dirichlet distributions.

Findings

Provides a multivariate normal integral expression for joint survival functions.

Establishes a relationship between multinomial survival functions and Dirichlet distributions.

Offers two proofs: expansion of Dirichlet density and Laplace's method.

Abstract

This paper presents a multivariate normal integral expression for the joint survival function of the cumulated components of any multinomial random vector. This result can be viewed as a multivariate analog of Equation (7) from Carter & Pollard (2004), who improved Tusn\'ady's inequality. Our findings are based on a crucial relationship between the joint survival function of the cumulated components of any multinomial random vector and the joint cumulative distribution function of a corresponding Dirichlet distribution. We offer two distinct proofs: the first expands the logarithm of the Dirichlet density, while the second employs Laplace's method applied to the Dirichlet integral.