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.

翻译：本文提出了任意多项随机向量累积分量联合生存函数的一个多元正态积分表达式。该结果可视为Carter & Pollard (2004)改进Tusnády不等式的公式(7)的多元推广。我们的发现基于任意多项随机向量累积分量的联合生存函数与对应狄利克雷分布联合累积分布函数之间的关键关系。我们提供了两种不同的证明方法：第一种通过展开狄利克雷分布密度的对数，第二种则对狄利克雷积分应用拉普拉斯方法。