Let $G$ be a locally compact abelian group, let $\lambda_G$ be Haar measure on $G$, and let $\hG$ denote the dual group of $G$. For $f\in L^1(G)$ and $\chi\in\hG$, let $\hf(\chi)$ be the Fourier transform of $f$. The Beurling--Gelfand spectral radius formula asserts then that
\[\left\lVertf^n\right\rVert_1^{1/n}\longrightarrow\max\limits_{\chi\in\hG}\bigl\lvert\hf(\chi)\bigr\rvert \]
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.