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...
| Main Authors: | , |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
Institute of Mathematical Statistics
2024
|
| Summary: | 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. |
|---|