Law of the iterated logarithm for random walks on nilpotent groups (Q5951611)

Let \((Z^j)\) be a sequence of independent, identically distributed random variables taking values on a simply connected, gradued, nilpotent Lie group \(G\). Consider the product \(T^n= Z^1\circ Z^2\circ\cdots\circ Z^n\). The authors show a law of the iterated logarithm for the random walk \((T^n)\) by providing the set of its limit points whenever it is suitably normalized by means of the sequences \((\sqrt{n\ln\ln n})\). The crucial idea to prove their theorem is to represent \(D_{1/\sqrt{n\ln\ln n}}T^n\) as the value at time \(1\) of the solution of a suitable ordinary differential equation with stochastic coefficients whose randomness is driven by a normalized polygon constructed by means of the sequence \((Z^j)\). This is done in the second part of the paper. In the first part of the paper, they consider this more general process and they prove a functional law of iterated logarithm. Finally, they consider the Heisenberg group, they compare their result with that proved by Crepel and Roynette.