Lindeberg–Feller theorems on Lie groups

`=1 ∫ |x|>ε |x| μn`(dx) = 0 for all ε > 0, where x 7→ |x| denotes a fixed homogeneous norm on the Heisenberg group. Professor E. Siebert suggested a way to prove similar results for a Lie group G. The main step is the generalization of Lemma 1 in Pap [4] for an arbitrary Lie group G, which gives an estimation for the Fourier transform of a probability measure on G in terms of integrals of local coordinates and a Hunt function, see Section 3. This lemma makes it possible to apply the accompanying Poisson system of a triangular system (μn`)`=1,...,kn;n>1 of probability measures on G in the usual way, see Section 4. Section 5 is devoted to Lindeberg–Feller type theorems on an arbitrary Lie group. Section 6 contains the important special case of stratified Lie groups, where the usual form of the Lindeberg–Feller theorem can be obtained.