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

Frédéric Ouimet

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 Tusnády'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.

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

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 Tusnády'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.

Paper Structure

This paper contains 3 sections, 1 theorem, 38 equations.

Table of Contents

  1. Introduction
  2. Quantile coupling
  3. Proofs

Key Result

Theorem \oldthetheorem

For all $\boldsymbol{k}\in \mathbb{N}_0^d \cap n \mathcal{S}_d$, one has Useful alternative expressions for $\widetilde{\gamma}(\boldsymbol{\varepsilon})$ and $\gamma^{\star}(\boldsymbol{s})$ can be found in eq:gamma.epsilon.alternative and eq:gamma.star.alternative, respectively.

Theorems & Definitions (5)

  • Theorem \oldthetheorem
  • Remark 1
  • proof : First proof of Theorem \ref{['thm:quantile.coupling']}
  • proof : Second proof of Theorem \ref{['thm:quantile.coupling']}
  • Remark 2