The central limit theorem on nilpotent Lie groups

We formulate and establish the central limit theorem for products of i.i.d. random variables on arbitrary simply connected nilpotent Lie groups, allowing a possible bias. We find that some interesting new phenomena arise in the presence of a bias: the walk spreads out at a higher rate in the ambient group, while the limiting hypoelliptic diffusion process may not always have full support. We use elementary Fourier analysis to establish our results, which include Berry–Esseen bounds under optimal moment assumptions as well as an analogue of Donsker’s invariance principle. Various examples of nilpotent Lie groups are treated in detail showing the variety of different behaviours. We also obtain a characterization of when the limiting distribution is an ordinary Gaussian and answer a question of Tutubalin regarding asymptotically close distributions on nilpotent Lie groups.