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：一个无限可交换随机变量序列总可被表示为独立同分布（i.i.d.）随机变量序列的混合。本文考虑可交换性的一种加权推广形式，允许通过权重函数沿序列修改单个随机变量的分布，前提是在这些权重函数的作用下仍存在某种共同的可交换基测度。我们研究加权可交换序列存在德·菲内蒂型表示的条件——该表示以满足加权形式独立同分布性质的分布的混合形式呈现。本文方法建立了一个嵌套的条件族，由此还可推导出其他经典结论的加权扩展形式，特别是零一律与大数定律的推广。