A bivariate random vector can exhibit either asymptotic independence or dependence between the largest values of its components. When used as a statistical model for risk assessment in fields such as finance, insurance or meteorology, it is crucial to understand which of the two asymptotic regimes occurs. Motivated by their ubiquity and flexibility, we consider the extremal dependence properties of vectors with a random scale construction (X₁,X₂) = R(W₁,W₂), with non-degenerate R > 0 independent of (W₁,W₂). Focusing on the presence and strength of asymptotic tail dependence, as expressed through commonly-used summary parameters, broad factors that affect the results are: the heaviness of the tails of R and (W₁,W₂), the shape of the support of (W₁,W₂), and dependence between (W₁,W₂). When R is distinctly lighter tailed than (W₁,W₂), the extremal dependence of (X₁,X₂) is typically the same as that of (W₁,W₂), whereas similar or heavier tails for R compared to (W₁,W₂) typically result in increased extremal dependence. Similar tail heavinesses represent the most interesting and technical cases, and we find both asymptotic independence and dependence of (X₁,X₂) possible in such cases when (W₁,W₂) exhibit asymptotic independence. The bivariate case often directly extends to higher-dimensional vectors and spatial processes, where the dependence is mainly analyzed in terms of summaries of bivariate sub-vectors. The results unify and extend many existing examples, and we use them to propose new models that encompass both dependence classes.

中文翻译：

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极端尺度的随机尺度构造

二元随机向量可以表现出渐近独立性，也可以表现出其成分最大值之间的依赖性。在金融，保险或气象学等领域用作风险评估的统计模型时，至关重要的是要了解这两种渐近状态中的哪一种发生。基于它们的普遍性和灵活性，我们考虑具有随机尺度构造（X ₁，X ₂）= R（W ₁，W ₂）且非简并R > 0独立于（W ₁，w ^ ₂）。通过常用的摘要参数表示，着眼于渐近尾部依赖的存在和强度，影响结果的广泛因素包括：R和（W ₁，W ₂）的尾部的沉重度，R的尾部支撑形状。 （W ₁，W ₂）以及（W ₁，W ₂）之间的依赖性。当R的尾部比（W ₁，W ₂）明显轻时，（X ₁，X ₂）的极值依赖性通常与（W ₁，W ₂），而与（W ₁，W ₂）相比，R的相似或较重的尾巴通常会导致极值依赖性增加。相似的尾部重量代表最有趣和技术性的情况，并且当（W ₁，W _2）时，我们发现（X ₁，X ₂）的渐近独立性和依赖性）表现出渐近独立性。双变量情况通常直接扩展到高维向量和空间过程，其中主要根据双变量子向量的摘要来分析相关性。结果统一并扩展了许多现有示例，我们使用它们来提出包含两个依赖关系类的新模型。