Title: The empirical discrete copula process Show Chinese title

Title: 经验离散 copula 过程

Abstract: This paper develops a general inferential framework for discrete copulas on finite supports in any dimension. The copula of a multivariate discrete distribution is defined as Csiszar's I-projection (i.e., the minimum-Kullback-Leibler divergence projection) of its joint probability array onto the polytope of uniform-margins probability arrays of the same size, and its empirical estimator is obtained by applying that same projection to the array of empirical frequencies observed on the sample. Under the assumption of random sampling, strong consistency and root-n-asymptotic normality of the empirical copula array is established, with an explicit "sandwich" form for its covariance. The theory is illustrated by deriving the large-sample distribution of Yule's concordance coefficient (the natural analogue of Spearman's rho for bivariate discrete distributions) and by constructing a test for quasi-independence in multivariate contingency tables. Our results not only complete the foundations of discrete-copula inference but also connect directly to entropically regularised optimal transport and other minimum-divergence problems. Show Chinese abstract

Abstract: 本文发展了一种通用的推断框架，用于任意维度上有限支撑上的离散copula。 多变量离散分布的copula被定义为其联合概率数组在具有相同尺寸的均匀边缘概率数组多面体上的Csiszar I-投影（即最小Kullback-Leibler散度投影），其经验估计量通过将相同的投影应用于样本中观察到的经验频率数组获得。 在随机抽样的假设下，建立了经验copula数组的强一致性和根号n渐近正态性，并给出了其协方差的显式“三明治”形式。 该理论通过导出Yule一致性系数（双变量离散分布中Spearman相关系数的自然类比）的大样本分布以及构建多元列联表中的准独立性检验得到了说明。 我们的结果不仅完善了离散copula推断的基础，还直接与熵正则化最优传输以及其他最小发散问题联系起来。