  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.
