Some highlights from the theory of multivariate symmetries

We explain how invariance in distribution under separate or joint contractions, permutations, or rotations can be defined in a natural way for d-dimensional arrays of random variables. In each case, the distribution is characterized by a general representation formula, often easy to state but surprisingly complicated to prove. Comparing the representations in the first two cases, one sees that an array on a tetrahedral index set is contractable iff it admits an extension to a jointly exchangeable array on the full rectangular index set. Multivariate rotatability is defined most naturally for continuous linear random functionals on tensor products of Hilbert spaces. Here the simplest examples are the multiple Wiener–Itô integrals, which also form the basic building blocks of the general representations. The rotatable theory can be used to derive similar representations for separately or jointly exchangeable or contractable random sheets. The present paper provides a non-technical survey of the mentioned results, the complete proofs being available elsewhere. We conclude with a list of open problems. 1 – Basic symmetries and classical results Many basic ideas in the area of probabilistic symmetries can be traced back to the pioneering work of Bruno de Finetti. After establishing, in 1930–37, his celebrated representation theorem for exchangeable sequences, he proposed in de Finetti (1938) the study of partial exchangeability of a random sequence, in the sense of invariance in distribution under a proper subgroup of permutations