Convergence of noncommutative triangular arrays of probability measures on a Lie group by H. Heyer and G. Pap Abstract. A measure–theoretic approach to the central limit problem for non- commutative infinitesimal arrays of random variables taking values in a Lie group G is given. Starting with an array {µ nℓ :(n, ℓ) ∈ N 2 } of probability mea- sures on G and instants 0  s  t one forms the finite convolution products µ n (s, t) := µ n,kn(s)+1 *···* µ n,kn(t) . The authors establish sufficient conditions in terms of L´ evy–Hunt characteristics for the sequence {µ n (s, t): n ∈ N} to converge towards a convolution hemigroup (generalized semigroup) of measures on G which turns out to be of bounded variation. In particularly, 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 differentiable functions and on the solution of weak backward evolution equations on G. Key words: Central limit theorem for Lie groups, noncommutative infinitesi- mal arrays of probability measures, convolution hemigroups, diffusion hemigroups, processes with independent increments. 1. Introduction The central limit problem of probability theory on a finite dimensional real Lie group G can be viewed as the study of convergence of infinitesimal triangular arrays of probability measures on G towards convolution hemigroups. More precisely one considers an array {µ nℓ :(n,ℓ) ∈ N 2 } of probability measures on G which for a sequence {k n : n ∈ N} of increasing c`ad functions k n : R + → Z + with jumps equal to 1 and satisfying k n (0) = 0 fulfills the infinitesimality condition lim n→∞ max{µ nℓ (∁U ):1  ℓ  k n (t)} =0 valid for all Borel neighbourhoods U of the neutral element e of G and for all t ∈ R + , one forms the convolution products µ n (s,t) := * kn(t) ℓ=kn(s)+1 µ nℓ 1