The central limit theorem for exchangeable random variables without moments
The paper deals with the question of existence of sequences \(a_ n\), \(0<b_ n\to \infty\) such that \[ (*)\quad b_ n^{-1}(X_ 1+X_ 2+...+X_ n-a_ n)\to N(0,1), \] where \(\{X_ n: n\geq 1\}\) is a sequence of exchangeable r.v.'s. The authors have proved that (*) implies that \(b_ n/n^{\alpha}\) must be slowly varying with \(\alpha =1/2\) or 1. Also necessary and sufficient conditions for (*) have been found.
scientific article; zbMATH DE number 4005236