  We introduce a new method for proving central limit theorems for random walk
on nilpotent groups. The method is illustrated in a local central limit theorem
on the Heisenberg group, weakening the necessary conditions on the driving
measure. As a second illustration, the method is used to study walks on the
$n\times n$ uni-upper triangular group with entries taken modulo $p$. The
method allows sharp answers to the behavior of individual coordinates:
coordinates immediately above the diagonal require order $p^2$ steps for
randomness, coordinates on the second diagonal require order $p$ steps;
coordinates on the $k$th diagonal require order $p^{\frac{2}{k}}$ steps.
