Itselfexchangeable

Itselfexchangeable is a term used in probability theory to describe objects whose joint distribution is invariant under permutations of their own indices or components. For a finite sequence X1, X2, …, Xn, the sequence is itselfexchangeable if for every permutation π of {1, …, n}, the random vector (X1, …, Xn) has the same distribution as (Xπ(1), …, Xπ(n)).

In infinite sequences, this condition aligns with the standard notion of exchangeability: the joint distribution of any finite subset of indices is invariant under all permutations of those indices. The phrase “itselfexchangeable” is sometimes used to emphasize invariance relative to the object’s own indexing, as opposed to invariance under external symmetries or operations.

Relation to known results: By de Finetti’s theorem, an infinite exchangeable sequence can be represented as conditionally i.i.d. given a latent random measure. This provides a structural characterization of itselfexchangeable sequences, highlighting that their randomness can be decomposed into a mixture of i.i.d. behaviors.

Examples and scope: Independent and identically distributed sequences are trivially exchangeable and hence itselfexchangeable. Finite samples drawn without replacement from a finite population also yield exchangeable joint distributions. More generally, many symmetric stochastic processes and arrays exhibit exchangeability under permutations of indices, supporting the idea captured by the term.

Limitations and usage: The concept of itselfexchangeable relies on a symmetry that may not hold in processes with time order, causality, or directional dependence. In practice, the term is not widely formalized beyond standard exchangeability, but it can be useful in discussions that stress permutation invariance within the object’s own structure.