\[\left\lVertf^n\right\rVert_1^{1/n}\longrightarrow\max\limits_{\chi\in\hG}\bigl\lvert\hf(\chi)\bigr\rvert \]

as $n\longrightarrow\infty$, where $f^n$ is
$n$-fold convolution of $f$ with itself. We use the analogue of this
formula for arbitrary (not necessarily abelian ) compact groups for an
approach to certain limit theorems for random walks. In particular, we
derive two well known results, appearing in Kloss and Shlosman
respectively: the first concerning convergence in the total variation
norm of $\mu^n$ to Haar measure, when $\mu$ is absolutely continuous,
and the second concerning the \emph{uniform} convergence of the
densities of the $\mu^n$, when $\mu$ has an $L^{1+\delta}$-density.