Central Limit Theorems for Exchangeable Random Variables When Limits Are Scale Mixtures of Normals

Abstract

Central limit theorems for exchangeable random variables are studied when limits are scale mixtures of normals. First, necessary and sufficient conditions are given under the asymptotic tail probability condition for the mixands: $$nP^\omega \left\{ {\left| {\xi _1 } \right| >\varepsilon b_n } \right\}\xrightarrow{P}0.$$ Second, when the weak limits have a particular form, i.e., the mixing measure comes directly from de Finetti's Theorem, necessary and sufficient conditions are given. Finally, some applications are discussed.