Central limit theorem with exchangeable summands and mixtures of stable laws as limits (Q2869748)

Let \(\left\{ X_{ij}:i,j=1,2,\dots\right\} \) be an array of exchangeable random variables. Exchangeability means that the joint distribution of every finite subset of \(m\) of these random variables depends only on \(m\) and not on the particular subset, \(m\geq1\). In this paper, the problem of convergence in law of normalized sums \(\mathbf{S}_{n}=\left( S_{1n}^{\ast} ,S_{2n}^{\ast},\dots\right) \) where NEWLINE\[NEWLINES_{in}^{\ast}=\frac{\sum_{j=1}^{n} X_{ij}-a_{n}}{b_{n}}NEWLINE\]NEWLINE and \(\left( a_{n}\right) _{n\geq1}\), \(\left( b_{n}\right) _{n\geq1}\) are sequences of real numbers such that \(b_{n}>0\), \(b_{n}\rightarrow\infty\) is examined. The authors give a partial characterization of the class of limiting laws of \(\mathbf{S}_{n}\) and they provide a criterion (necessary and sufficient conditions) for the weak convergence of \(\mathbf{S}_{n}\) to a specific form chosen from the class (mixtures of Gaussian distribution, mixtures of stable laws). Sufficent conditions for convergence of sums in a single row are also proved.