De Finetti's theorem, also called the de Finetti-Hewitt-Savage theorem, is a foundational result in probability and statistics. Roughly, it says that an infinite sequence of exchangeable random variables can always be written as a mixture of independent and identically distributed (i.i.d.) sequences of random variables. In this paper, we consider a weighted generalization of exchangeability that allows for weight functions to modify the individual distributions of the random variables along the sequence, provided that -- modulo these weight functions -- there is still some common exchangeable base measure. We study conditions under which a de Finetti-type representation exists for weighted exchangeable sequences, as a mixture of distributions which satisfy a weighted form of the i.i.d. property. Our approach establishes a nested family of conditions that lead to weighted extensions of other well-known related results as well, in particular, extensions of the zero-one law and the law of large numbers.

翻译：德菲内蒂定理（亦称德菲内蒂-休伊特-萨维奇定理）是概率论与统计学中的基础性结果。该定理大致指出：无限可交换随机变量序列总能表示为独立同分布随机变量序列的混合分布。本文考虑可交换性的加权泛化形式：允许通过权函数沿序列方向调整各随机变量的个体分布，但要求在这些权函数作用下，仍存在共同的、可交换的基底测度。我们系统研究了加权可交换序列存在德菲内蒂型表示的条件，该表示将序列刻画为满足独立同分布性质的加权形式的混合分布。我们的方法建立了一个嵌套条件族，该条件族同样能导出其他经典结论的加权扩展，特别是零一律和大数定律的推广形式。