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（也称为德菲内蒂-休伊特-萨维奇定理）是概率论与统计学中的基础性成果。其核心思想是：无穷可交换随机变量序列总可表示为独立同分布（i.i.d.）随机变量序列的混合分布。本文研究可交换性的加权推广形式，该形式允许通过权函数沿序列修改随机变量的个体分布，前提是在这些权函数作用下仍存在某种公共的可交换基测度。我们考察了加权可交换序列存在德菲内蒂型表示的条件，该表示可表示为满足加权独立同分布性质的分布的混合。本文方法建立了嵌套条件族，由此还可推广其他经典相关结论的加权形式，特别是0-1律和大数定律的扩展。