We study the impact of dependence uncertainty on the expectation of the product of $d$ random variables, $\mathbb{E}(X_1X_2\cdots X_d)$ when $X_i \sim F_i$ for all~$i$. Under some conditions on the $F_i$, explicit sharp bounds are obtained and a numerical method is provided to approximate them for arbitrary choices of the $F_i$. The results are applied to assess the impact of dependence uncertainty on coskewness. In this regard, we introduce a novel notion of "standardized rank coskewness," which is invariant under strictly increasing transformations and takes values in $[-1,\ 1]$.

翻译：我们研究相依性不确定性对乘积期望值 $\mathbb{E}(X_1X_2\cdots X_d)$ 的影响，其中每个随机变量 $X_i$ 服从边际分布 $F_i$。在 $F_i$ 满足特定条件时，给出了显式的紧界；对于任意 $F_i$ 选择，则提供数值逼近方法。我们将这些结果应用于评估相依性不确定性对协偏度的影响，并由此引入一种新型的"标准化秩协偏度"概念——该指标在严格递增变换下保持不变，且取值于 $[-1,\ 1]$ 区间。