A central limit problem for partially exchangeable random variables

The present paper deals with the central limit problem for $((S_{1n},S_{2n},\ldots))_{n}$ when $S_{in}=\sum_{j=1}^n\xi_{ij}^{(n)}$ $(i=1,2,\ldots)$ and, for every $n$, $\{\xi_{ij}^{(n)}\: i=1,2,\ldots;j=1,\ldots,n\}$ is an array of partially exchangeable random variables. It is shown that, under suitable "negligibility" conditions, the class of limiting laws coincides with that of all exchangeable laws which are presentable as mixtures of infinitely divisible distributions. Moreover, necessary and sufficient conditions for convergence to any specified element of that class are provided. Criteria for three remarkable limit types (mixture of Gaussian, Poisson, degenerate probability distributions) are explained. It is also proved that the class of limiting laws can be characterized in terms of mixtures of stable laws, when $\xi_{ij}^{(n)}=X_{ij}/a_n$ $(a_n\rightarrow +\infty)$ and the $X_{ij}$'s $(i,j=1,2,\ldots)$ are assumed to be exchangeable. Finally, one shows that a few basic, well-known central limit theorems for sequences of exchangeable random variables can be obtained as simple corollaries of the main results proved in the present paper.