Large Deviations for Exchangeable Random Vectors

Say that a family _θⁿ: θ ∈ Θ\ of sequences of probability measures is exponentially continuous if whenever θₙ → θ, the sequence θₙⁿ\ satisfies a large deviation principle with rate function λ_θ. If Θ is compact and _θⁿ\ is exponentially continuous, then the mixture Pⁿ(A) =: ∫_Θ P_θⁿ(A)dμ(θ) satisfies a large deviation principle with rate function λ(x) =: inf\λ_θ(x): θ ∈ S(μ)\, where S(μ) is the support of the mixing measure μ. If X₁,X₂,… is a sequence of i.i.d. random vectors, ̄ₙ\ the corresponding sequence of sample means and P_θⁿ =: P_θ̄⁻¹ₙ, then _θⁿ\ is exponentially continuous if the classical rate function λ_θ(ν) is jointly lower semicontinuous and a uniform integrability condition introduced by de Acosta is satisfied. These results are applied in Section 4 to derive a large deviation theory for exchangeable random variables; the resulting rate functions are typically nonconvex. If the parameter space Θ is not compact, then examples can be constructed where a full large deviation principle is not satisfied because the upper bound fails for a noncompact set.