ar X iv : m at h / 04 06 57 8 v 4 [ m at h . D S ] 6 S ep 2 00 5 EXCHANGEABLE MEASURES FOR SUBSHIFTS

Let Ω be a Borel subset of S N where S is countable. A measure is called exchangeable on Ω, if it is supported on Ω and is invariant under every Borel automorphism of Ω which permutes at most finitely many coordinates. De-Finetti's theorem characterizes these measures when Ω = S N. We apply the ergodic theory of equivalence relations to study the case Ω = S N , and obtain versions of this theorem when Ω is a countable state Markov shift, and when Ω is the collection of beta expansions of real numbers in [0, 1] (a non-Markovian constraint). §0 Introduction Exchangeability. De-Finetti's theorem says that if a stochastic process {X n } n≥1 is exchangeable, i.e. all finite permutations {X π(n) } of {X n } n≥1 are distributed like {X n }, then it is distributed as a mixture of i.i.d. distributions. Here is a seemingly stronger, but equivalent formulation: Let K be the collection of all bi-measurable bijections κ : A → B (A, B ⊆ S N Borel) for which for every x κ(x) is some finite permutation 1 of x; then any Borel probability measure m on Ω := S N such that m • κ| Dom (κ) = m| Dom (κ) for all κ ∈ K is an average of Bernoulli measures. De-Finetti's theorem is instrumental in statistical modeling of sequential sampling , because it determines the form of joint distributions whenever the sampling order is unimportant. But sometimes the sampling order is subject to non-permutation invariant deterministic constraints. In these cases the joint distribution cannot be assumed to be exchangeable. Nevertheless, one can still ask for 'the most exchangeable' compatible distributions. There are various ways to formalize this. In this paper we use the following: Let Ω be a Borel subset of S N (thought of as the space of realizations of {X n } n≥1 subject to a collection of deterministic constraints), and set K(Ω) := {κ ∈ K : Dom (κ), Im (κ) ⊆ Ω}. A Borel measure m on Ω is called exchangeable on Ω if m • κ| Dom (κ) = m| Dom (κ) for all κ ∈ K(Ω). When Ω = S N , this reduces to the usual notion of exchangeability. This definition of exchangeability is the one used by Petersen & Schmidt in the context of finite state Markov shifts [Pe-S], but is not …