Convergence of Noncommutative Triangular Arrays of Probability Measures on a Lie Group

A measure-theoretic approach to the central limit problem for noncommutative infinitesimal arrays of random variables taking values in a Lie group G is given. Starting with an array of probability measures on G and instance 0st one forms the finite convolution products . The authors establish sufficient conditions in terms of Lévy-Hunt characteristics for the sequence to converge towards a convolution hemigroup (generalized semigroup) of measures on G which turns out to be of bounded variation. In particular, conditions are stated that force the limiting hemigroup to be a diffusion hemigroup. The method applied in the proofs is based on properly chosen spaces of difierentiable functions and on the solution of weak backward evolution equations on G.