We introduce the concept of a product disintegration, which generalizes exchangeability by allowing one to render conditional independence in terms of a suitable hidden sequence of random probabilities. We prove that, given a sequence of random elements in a compact metric space, a strong law of large numbers (SLLN) holds for observables of this process if and only if the process admits a product disintegration that itself satisfies a type of SLLN. As an application, we provide a characterization of the SLLN for identically distributed (not necessarily independent) Bernoulli sequences.

中文翻译：

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产品分解：通过条件独立的大数定律

我们引入了产品分解的概念，它通过允许人们根据适当的随机概率隐藏序列呈现条件独立性来概括可交换性。我们证明，给定紧凑度量空间中的随机元素序列，当且仅当该过程允许产品分解且其本身满足某种类型的 SLLN 时，强大数定律 (SLLN) 才适用于该过程的可观测量。作为一个应用，我们提供了相同分布（不一定独立）伯努利序列的 SLLN 特征。