On the central limit problem for partially exchangeable random variables with values in a Hilbert space

This paper deals with a central limit problem for columnwise exchangeable arrays of a Hilbert space valued random variables satisfying the condition of conditional uniform asymptotic negligibility. It is proved that rowwise sums may converge in distribution only to an exchangeable sequence of random variables whose distribution is a mixture of infinitely divisible distributions. Conditions for convergence to a specific distribution and, in particular, to a mixture of Gaussian distributions, are also given.