A Very General De Finetti-Type Theorem

A few years ago it turned out that De Finetti’s famous theorem concerning exchangeable 0–1 valued random variables can also be proved by harmonic analysis means, applied to the special semigroup {(k,n) ∊ IN o 2 |k ≦ n}. This is no pure coincidence; a careful inspection of the new proof revealed that many other De Finetti-type theorems, old and new ones, could be shown the same way, among them Schoenberg’s representation of spherically symmetric random sequences, Hewitt and Savage’s far-reaching generalisation of De Finetti’s original result, and numerous characterisations of mixtures of i.i.d.-sequences with concrete prescribed distributions.