Diffusion approximations for random walks on nilpotent Lie groups (Q1304063)

Let \({Z^{k}}\) be a sequence of i.i.d. \(\mathbb{R}^{d}\)-valued random variables. For every \(n\in\mathbb{N}\), set \[ S_{t}^{n} = \sum_{k=1}^{[nt]} Z^{k} + (nt -[nt])Z[nt]+1). \] \(S^{n}\) are called polygonals. Consider a differential equation \(dY_t^{n} = b^{n}(Y_t^{n})dt + \sigma^{n}(Y_t^{n})\frac{1}{\sqrt n} S_{t}^{n} dt\). Then it is shown that under some assumptions, its solutions are weakly convergent in \(C([0,T],\mathbb{R}^d)\) to solutions of a stochastic differential equation \(dY_{t} = \widetilde{b}(Y_t)dt + \sigma( Y_{t}) dW_{t}\), where \(W\) is an \(m\)-dimensional Brownian process. These are applied to simply connected graded nilpotent group valued random sequences, and on this group, a functional form of the central limit theorem is given. As an example, Heisenberg group is treated.